# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1553012091 0
# Node ID 4a9167f377b0633e4714726cc36c5075294ff195
# Parent  eddcc7c726f3e57e73c59d328e19a3455874f457
new material about topology, etc.; also fixes for yesterday's

diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Abstract_Topology.thy
--- a/src/HOL/Analysis/Abstract_Topology.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Abstract_Topology.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -8,7 +8,6 @@
     Complex_Main
     "HOL-Library.Set_Idioms"
     "HOL-Library.FuncSet"
-    (* Path_Connected *)
 begin
 
 subsection \<open>General notion of a topology as a value\<close>
@@ -427,7 +426,7 @@
 lemma topspace_euclidean [simp]: "topspace euclidean = UNIV"
   by (force simp: topspace_def)
 
-lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
+lemma topspace_euclidean_subtopology[simp]: "topspace (top_of_set S) = S"
   by (simp add: topspace_subtopology)
 
 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
@@ -435,7 +434,7 @@
 
 declare closed_closedin [symmetric, simp]
 
-lemma openin_subtopology_self [simp]: "openin (subtopology euclidean S) S"
+lemma openin_subtopology_self [simp]: "openin (top_of_set S) S"
   by (metis openin_topspace topspace_euclidean_subtopology)
 
 subsubsection\<open>The most basic facts about the usual topology and metric on R\<close>
@@ -445,51 +444,51 @@
 
 subsection \<open>Basic "localization" results are handy for connectedness.\<close>
 
-lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
+lemma openin_open: "openin (top_of_set U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
   by (auto simp: openin_subtopology)
 
 lemma openin_Int_open:
-   "\<lbrakk>openin (subtopology euclidean U) S; open T\<rbrakk>
-        \<Longrightarrow> openin (subtopology euclidean U) (S \<inter> T)"
+   "\<lbrakk>openin (top_of_set U) S; open T\<rbrakk>
+        \<Longrightarrow> openin (top_of_set U) (S \<inter> T)"
 by (metis open_Int Int_assoc openin_open)
 
-lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
+lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (top_of_set U) (U \<inter> S)"
   by (auto simp: openin_open)
 
 lemma open_openin_trans[trans]:
-  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
+  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (top_of_set S) T"
   by (metis Int_absorb1  openin_open_Int)
 
-lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
+lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (top_of_set T) S"
   by (auto simp: openin_open)
 
-lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
+lemma closedin_closed: "closedin (top_of_set U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
   by (simp add: closedin_subtopology Int_ac)
 
-lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)"
+lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (top_of_set U) (U \<inter> S)"
   by (metis closedin_closed)
 
-lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
+lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (top_of_set T) S"
   by (auto simp: closedin_closed)
 
 lemma closedin_closed_subset:
- "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
-             \<Longrightarrow> closedin (subtopology euclidean T) S"
+ "\<lbrakk>closedin (top_of_set U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
+             \<Longrightarrow> closedin (top_of_set T) S"
   by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)
 
 lemma finite_imp_closedin:
   fixes S :: "'a::t1_space set"
-  shows "\<lbrakk>finite S; S \<subseteq> T\<rbrakk> \<Longrightarrow> closedin (subtopology euclidean T) S"
+  shows "\<lbrakk>finite S; S \<subseteq> T\<rbrakk> \<Longrightarrow> closedin (top_of_set T) S"
     by (simp add: finite_imp_closed closed_subset)
 
 lemma closedin_singleton [simp]:
   fixes a :: "'a::t1_space"
-  shows "closedin (subtopology euclidean U) {a} \<longleftrightarrow> a \<in> U"
+  shows "closedin (top_of_set U) {a} \<longleftrightarrow> a \<in> U"
 using closedin_subset  by (force intro: closed_subset)
 
 lemma openin_euclidean_subtopology_iff:
   fixes S U :: "'a::metric_space set"
-  shows "openin (subtopology euclidean U) S \<longleftrightarrow>
+  shows "openin (top_of_set U) S \<longleftrightarrow>
     S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof
@@ -513,16 +512,16 @@
 
 lemma connected_openin:
       "connected S \<longleftrightarrow>
-       \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
-                 openin (subtopology euclidean S) E2 \<and>
+       \<not>(\<exists>E1 E2. openin (top_of_set S) E1 \<and>
+                 openin (top_of_set S) E2 \<and>
                  S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
   apply (simp add: connected_def openin_open disjoint_iff_not_equal, safe)
   by (simp_all, blast+)  (* SLOW *)
 
 lemma connected_openin_eq:
       "connected S \<longleftrightarrow>
-       \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
-                 openin (subtopology euclidean S) E2 \<and>
+       \<not>(\<exists>E1 E2. openin (top_of_set S) E1 \<and>
+                 openin (top_of_set S) E2 \<and>
                  E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
                  E1 \<noteq> {} \<and> E2 \<noteq> {})"
   apply (simp add: connected_openin, safe, blast)
@@ -531,8 +530,8 @@
 lemma connected_closedin:
       "connected S \<longleftrightarrow>
        (\<nexists>E1 E2.
-        closedin (subtopology euclidean S) E1 \<and>
-        closedin (subtopology euclidean S) E2 \<and>
+        closedin (top_of_set S) E1 \<and>
+        closedin (top_of_set S) E2 \<and>
         S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
        (is "?lhs = ?rhs")
 proof
@@ -561,8 +560,8 @@
 lemma connected_closedin_eq:
       "connected S \<longleftrightarrow>
            \<not>(\<exists>E1 E2.
-                 closedin (subtopology euclidean S) E1 \<and>
-                 closedin (subtopology euclidean S) E2 \<and>
+                 closedin (top_of_set S) E1 \<and>
+                 closedin (top_of_set S) E2 \<and>
                  E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
                  E1 \<noteq> {} \<and> E2 \<noteq> {})"
   apply (simp add: connected_closedin, safe, blast)
@@ -571,26 +570,26 @@
 text \<open>These "transitivity" results are handy too\<close>
 
 lemma openin_trans[trans]:
-  "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow>
-    openin (subtopology euclidean U) S"
+  "openin (top_of_set T) S \<Longrightarrow> openin (top_of_set U) T \<Longrightarrow>
+    openin (top_of_set U) S"
   by (metis openin_Int_open openin_open)
 
-lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
+lemma openin_open_trans: "openin (top_of_set T) S \<Longrightarrow> open T \<Longrightarrow> open S"
   by (auto simp: openin_open intro: openin_trans)
 
 lemma closedin_trans[trans]:
-  "closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow>
-    closedin (subtopology euclidean U) S"
+  "closedin (top_of_set T) S \<Longrightarrow> closedin (top_of_set U) T \<Longrightarrow>
+    closedin (top_of_set U) S"
   by (auto simp: closedin_closed closed_Inter Int_assoc)
 
-lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
+lemma closedin_closed_trans: "closedin (top_of_set T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
   by (auto simp: closedin_closed intro: closedin_trans)
 
 lemma openin_subtopology_Int_subset:
-   "\<lbrakk>openin (subtopology euclidean u) (u \<inter> S); v \<subseteq> u\<rbrakk> \<Longrightarrow> openin (subtopology euclidean v) (v \<inter> S)"
+   "\<lbrakk>openin (top_of_set u) (u \<inter> S); v \<subseteq> u\<rbrakk> \<Longrightarrow> openin (top_of_set v) (v \<inter> S)"
   by (auto simp: openin_subtopology)
 
-lemma openin_open_eq: "open s \<Longrightarrow> (openin (subtopology euclidean s) t \<longleftrightarrow> open t \<and> t \<subseteq> s)"
+lemma openin_open_eq: "open s \<Longrightarrow> (openin (top_of_set s) t \<longleftrightarrow> open t \<and> t \<subseteq> s)"
   using open_subset openin_open_trans openin_subset by fastforce
 
 
@@ -1684,11 +1683,11 @@
   "continuous_map (discrete_topology U) X f \<longleftrightarrow> f ` U \<subseteq> topspace X"
   by (auto simp: continuous_map_def)
 
-lemma continuous_map_iff_continuous_real [simp]: "continuous_map (subtopology euclideanreal S) euclideanreal g = continuous_on S g"
-  by (force simp: continuous_map openin_subtopology continuous_on_open_invariant)
-
-lemma continuous_map_iff_continuous_real2 [simp]: "continuous_map euclideanreal euclideanreal g = continuous_on UNIV g"
-  by (metis continuous_map_iff_continuous_real subtopology_UNIV)
+lemma continuous_map_iff_continuous [simp]: "continuous_map (subtopology euclidean S) euclidean g = continuous_on S g"
+  by (fastforce simp add: continuous_map openin_subtopology continuous_on_open_invariant)
+
+lemma continuous_map_iff_continuous2 [simp]: "continuous_map euclidean euclidean g = continuous_on UNIV g"
+  by (metis continuous_map_iff_continuous subtopology_UNIV)
 
 lemma continuous_map_openin_preimage_eq:
    "continuous_map X Y f \<longleftrightarrow>
@@ -2844,9 +2843,8 @@
     done
 qed
 
-lemma connectedin_iff_connected_real [simp]:
-     "connectedin euclideanreal S \<longleftrightarrow> connected S"
-    by (simp add: connected_def connectedin)
+lemma connectedin_iff_connected [simp]: "connectedin euclidean S \<longleftrightarrow> connected S"
+  by (simp add: connected_def connectedin)
 
 lemma connectedin_closedin:
    "connectedin X S \<longleftrightarrow>
@@ -3190,7 +3188,7 @@
   unfolding compact_space_alt
   using openin_subset by fastforce
 
-lemma compactin_euclideanreal_iff [simp]: "compactin euclideanreal S \<longleftrightarrow> compact S"
+lemma compactin_euclidean_iff [simp]: "compactin euclidean S \<longleftrightarrow> compact S"
   by (simp add: compact_eq_Heine_Borel compactin_def) meson
 
 lemma compactin_absolute [simp]:
@@ -3639,36 +3637,127 @@
   unfolding embedding_map_def
   using homeomorphic_space by blast
 
+
+subsection\<open>Retraction and section maps\<close>
+
+definition retraction_maps :: "'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> bool"
+  where "retraction_maps X Y f g \<equiv>
+           continuous_map X Y f \<and> continuous_map Y X g \<and> (\<forall>x \<in> topspace Y. f(g x) = x)"
+
+definition section_map :: "'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
+  where "section_map X Y f \<equiv> \<exists>g. retraction_maps Y X g f"
+
+definition retraction_map :: "'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
+  where "retraction_map X Y f \<equiv> \<exists>g. retraction_maps X Y f g"
+
+lemma retraction_maps_eq:
+   "\<lbrakk>retraction_maps X Y f g; \<And>x. x \<in> topspace X \<Longrightarrow> f x = f' x; \<And>x. x \<in> topspace Y \<Longrightarrow> g x = g' x\<rbrakk>
+        \<Longrightarrow> retraction_maps X Y f' g'"
+  unfolding retraction_maps_def by (metis (no_types, lifting) continuous_map_def continuous_map_eq)
+
+lemma section_map_eq:
+   "\<lbrakk>section_map X Y f; \<And>x. x \<in> topspace X \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> section_map X Y g"
+  unfolding section_map_def using retraction_maps_eq by blast
+
+lemma retraction_map_eq:
+   "\<lbrakk>retraction_map X Y f; \<And>x. x \<in> topspace X \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> retraction_map X Y g"
+  unfolding retraction_map_def using retraction_maps_eq by blast
+
+lemma homeomorphic_imp_retraction_maps:
+   "homeomorphic_maps X Y f g \<Longrightarrow> retraction_maps X Y f g"
+  by (simp add: homeomorphic_maps_def retraction_maps_def)
+
+lemma section_and_retraction_eq_homeomorphic_map:
+   "section_map X Y f \<and> retraction_map X Y f \<longleftrightarrow> homeomorphic_map X Y f"
+  apply (auto simp: section_map_def retraction_map_def homeomorphic_map_maps retraction_maps_def homeomorphic_maps_def)
+  by (metis (full_types) continuous_map_image_subset_topspace image_subset_iff)
+
+lemma section_imp_embedding_map:
+   "section_map X Y f \<Longrightarrow> embedding_map X Y f"
+  unfolding section_map_def embedding_map_def homeomorphic_map_maps retraction_maps_def homeomorphic_maps_def
+  by (force simp: continuous_map_in_subtopology continuous_map_from_subtopology topspace_subtopology)
+
+lemma retraction_imp_quotient_map:
+  assumes "retraction_map X Y f"
+  shows "quotient_map X Y f"
+  unfolding quotient_map_def
+proof (intro conjI subsetI allI impI)
+  show "f ` topspace X = topspace Y"
+    using assms by (force simp: retraction_map_def retraction_maps_def continuous_map_def)
+next
+  fix U
+  assume U: "U \<subseteq> topspace Y"
+  have "openin Y U"
+    if "\<forall>x\<in>topspace Y. g x \<in> topspace X" "\<forall>x\<in>topspace Y. f (g x) = x"
+       "openin Y {x \<in> topspace Y. g x \<in> {x \<in> topspace X. f x \<in> U}}" for g
+    using openin_subopen U that by fastforce
+  then show "openin X {x \<in> topspace X. f x \<in> U} = openin Y U"
+    using assms by (auto simp: retraction_map_def retraction_maps_def continuous_map_def)
+qed
+
+lemma retraction_maps_compose:
+   "\<lbrakk>retraction_maps X Y f f'; retraction_maps Y Z g g'\<rbrakk> \<Longrightarrow> retraction_maps X Z (g \<circ> f) (f' \<circ> g')"
+  by (clarsimp simp: retraction_maps_def continuous_map_compose) (simp add: continuous_map_def)
+
+lemma retraction_map_compose:
+   "\<lbrakk>retraction_map X Y f; retraction_map Y Z g\<rbrakk> \<Longrightarrow> retraction_map X Z (g \<circ> f)"
+  by (meson retraction_map_def retraction_maps_compose)
+
+lemma section_map_compose:
+   "\<lbrakk>section_map X Y f; section_map Y Z g\<rbrakk> \<Longrightarrow> section_map X Z (g \<circ> f)"
+  by (meson retraction_maps_compose section_map_def)
+
+lemma surjective_section_eq_homeomorphic_map:
+   "section_map X Y f \<and> f ` (topspace X) = topspace Y \<longleftrightarrow> homeomorphic_map X Y f"
+  by (meson section_and_retraction_eq_homeomorphic_map section_imp_embedding_map surjective_embedding_map)
+
+lemma surjective_retraction_or_section_map:
+   "f ` (topspace X) = topspace Y \<Longrightarrow> retraction_map X Y f \<or> section_map X Y f \<longleftrightarrow> retraction_map X Y f"
+  using section_and_retraction_eq_homeomorphic_map surjective_section_eq_homeomorphic_map by fastforce
+
+lemma retraction_imp_surjective_map:
+   "retraction_map X Y f \<Longrightarrow> f ` (topspace X) = topspace Y"
+  by (simp add: retraction_imp_quotient_map quotient_imp_surjective_map)
+
+lemma section_imp_injective_map:
+   "\<lbrakk>section_map X Y f; x \<in> topspace X; y \<in> topspace X\<rbrakk> \<Longrightarrow> f x = f y \<longleftrightarrow> x = y"
+  by (metis (mono_tags, hide_lams) retraction_maps_def section_map_def)
+
+lemma retraction_maps_to_retract_maps:
+   "retraction_maps X Y r s
+        \<Longrightarrow> retraction_maps X (subtopology X (s ` (topspace Y))) (s \<circ> r) id"
+  unfolding retraction_maps_def
+  by (auto simp: continuous_map_compose continuous_map_into_subtopology continuous_map_from_subtopology)
 subsection \<open>Continuity\<close>
 
 lemma continuous_on_open:
   "continuous_on S f \<longleftrightarrow>
-    (\<forall>T. openin (subtopology euclidean (f ` S)) T \<longrightarrow>
-      openin (subtopology euclidean S) (S \<inter> f -` T))"
+    (\<forall>T. openin (top_of_set (f ` S)) T \<longrightarrow>
+      openin (top_of_set S) (S \<inter> f -` T))"
   unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute
   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
 
 lemma continuous_on_closed:
   "continuous_on S f \<longleftrightarrow>
-    (\<forall>T. closedin (subtopology euclidean (f ` S)) T \<longrightarrow>
-      closedin (subtopology euclidean S) (S \<inter> f -` T))"
+    (\<forall>T. closedin (top_of_set (f ` S)) T \<longrightarrow>
+      closedin (top_of_set S) (S \<inter> f -` T))"
   unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute
   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
 
 lemma continuous_on_imp_closedin:
-  assumes "continuous_on S f" "closedin (subtopology euclidean (f ` S)) T"
-  shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
+  assumes "continuous_on S f" "closedin (top_of_set (f ` S)) T"
+  shows "closedin (top_of_set S) (S \<inter> f -` T)"
   using assms continuous_on_closed by blast
 
 subsection%unimportant \<open>Half-global and completely global cases\<close>
 
 lemma continuous_openin_preimage_gen:
   assumes "continuous_on S f"  "open T"
-  shows "openin (subtopology euclidean S) (S \<inter> f -` T)"
+  shows "openin (top_of_set S) (S \<inter> f -` T)"
 proof -
   have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
     by auto
-  have "openin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
+  have "openin (top_of_set (f ` S)) (T \<inter> f ` S)"
     using openin_open_Int[of T "f ` S", OF assms(2)] unfolding openin_open by auto
   then show ?thesis
     using assms(1)[unfolded continuous_on_open, THEN spec[where x="T \<inter> f ` S"]]
@@ -3677,11 +3766,11 @@
 
 lemma continuous_closedin_preimage:
   assumes "continuous_on S f" and "closed T"
-  shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
+  shows "closedin (top_of_set S) (S \<inter> f -` T)"
 proof -
   have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
     by auto
-  have "closedin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
+  have "closedin (top_of_set (f ` S)) (T \<inter> f ` S)"
     using closedin_closed_Int[of T "f ` S", OF assms(2)]
     by (simp add: Int_commute)
   then show ?thesis
@@ -3691,7 +3780,7 @@
 
 lemma continuous_openin_preimage_eq:
    "continuous_on S f \<longleftrightarrow>
-    (\<forall>T. open T \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T))"
+    (\<forall>T. open T \<longrightarrow> openin (top_of_set S) (S \<inter> f -` T))"
 apply safe
 apply (simp add: continuous_openin_preimage_gen)
 apply (fastforce simp add: continuous_on_open openin_open)
@@ -3699,7 +3788,7 @@
 
 lemma continuous_closedin_preimage_eq:
    "continuous_on S f \<longleftrightarrow>
-    (\<forall>T. closed T \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` T))"
+    (\<forall>T. closed T \<longrightarrow> closedin (top_of_set S) (S \<inter> f -` T))"
 apply safe
 apply (simp add: continuous_closedin_preimage)
 apply (fastforce simp add: continuous_on_closed closedin_closed)
@@ -3733,9 +3822,9 @@
   by (simp add: closed_vimage continuous_on_eq_continuous_within)
 
 lemma Times_in_interior_subtopology:
-  assumes "(x, y) \<in> U" "openin (subtopology euclidean (S \<times> T)) U"
-  obtains V W where "openin (subtopology euclidean S) V" "x \<in> V"
-                    "openin (subtopology euclidean T) W" "y \<in> W" "(V \<times> W) \<subseteq> U"
+  assumes "(x, y) \<in> U" "openin (top_of_set (S \<times> T)) U"
+  obtains V W where "openin (top_of_set S) V" "x \<in> V"
+                    "openin (top_of_set T) W" "y \<in> W" "(V \<times> W) \<subseteq> U"
 proof -
   from assms obtain E where "open E" "U = S \<times> T \<inter> E" "(x, y) \<in> E" "x \<in> S" "y \<in> T"
     by (auto simp: openin_open)
@@ -3744,8 +3833,8 @@
     by blast
   show ?thesis
   proof
-    show "openin (subtopology euclidean S) (E1 \<inter> S)"
-      "openin (subtopology euclidean T) (E2 \<inter> T)"
+    show "openin (top_of_set S) (E1 \<inter> S)"
+      "openin (top_of_set T) (E2 \<inter> T)"
       using \<open>open E1\<close> \<open>open E2\<close>
       by (auto simp: openin_open)
     show "x \<in> E1 \<inter> S" "y \<in> E2 \<inter> T"
@@ -3757,20 +3846,20 @@
 qed
 
 lemma closedin_Times:
-  "closedin (subtopology euclidean S) S' \<Longrightarrow> closedin (subtopology euclidean T) T' \<Longrightarrow>
-    closedin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
+  "closedin (top_of_set S) S' \<Longrightarrow> closedin (top_of_set T) T' \<Longrightarrow>
+    closedin (top_of_set (S \<times> T)) (S' \<times> T')"
   unfolding closedin_closed using closed_Times by blast
 
 lemma openin_Times:
-  "openin (subtopology euclidean S) S' \<Longrightarrow> openin (subtopology euclidean T) T' \<Longrightarrow>
-    openin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
+  "openin (top_of_set S) S' \<Longrightarrow> openin (top_of_set T) T' \<Longrightarrow>
+    openin (top_of_set (S \<times> T)) (S' \<times> T')"
   unfolding openin_open using open_Times by blast
 
 lemma openin_Times_eq:
   fixes S :: "'a::topological_space set" and T :: "'b::topological_space set"
   shows
-    "openin (subtopology euclidean (S \<times> T)) (S' \<times> T') \<longleftrightarrow>
-      S' = {} \<or> T' = {} \<or> openin (subtopology euclidean S) S' \<and> openin (subtopology euclidean T) T'"
+    "openin (top_of_set (S \<times> T)) (S' \<times> T') \<longleftrightarrow>
+      S' = {} \<or> T' = {} \<or> openin (top_of_set S) S' \<and> openin (top_of_set T) T'"
     (is "?lhs = ?rhs")
 proof (cases "S' = {} \<or> T' = {}")
   case True
@@ -3782,13 +3871,13 @@
   show ?thesis
   proof
     assume ?lhs
-    have "openin (subtopology euclidean S) S'"
+    have "openin (top_of_set S) S'"
       apply (subst openin_subopen, clarify)
       apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
       using \<open>y \<in> T'\<close>
        apply auto
       done
-    moreover have "openin (subtopology euclidean T) T'"
+    moreover have "openin (top_of_set T) T'"
       apply (subst openin_subopen, clarify)
       apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
       using \<open>x \<in> S'\<close>
@@ -3805,7 +3894,7 @@
 
 lemma Lim_transform_within_openin:
   assumes f: "(f \<longlongrightarrow> l) (at a within T)"
-    and "openin (subtopology euclidean T) S" "a \<in> S"
+    and "openin (top_of_set T) S" "a \<in> S"
     and eq: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
   shows "(g \<longlongrightarrow> l) (at a within T)"
 proof -
@@ -3827,8 +3916,8 @@
 lemma continuous_on_open_gen:
   assumes "f ` S \<subseteq> T"
     shows "continuous_on S f \<longleftrightarrow>
-             (\<forall>U. openin (subtopology euclidean T) U
-                  \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U))"
+             (\<forall>U. openin (top_of_set T) U
+                  \<longrightarrow> openin (top_of_set S) (S \<inter> f -` U))"
      (is "?lhs = ?rhs")
 proof
   assume ?lhs
@@ -3841,23 +3930,23 @@
   proof (clarsimp simp add: continuous_openin_preimage_eq)
     fix U::"'a set"
     assume "open U"
-    then have "openin (subtopology euclidean S) (S \<inter> f -` (U \<inter> T))"
+    then have "openin (top_of_set S) (S \<inter> f -` (U \<inter> T))"
       by (metis R inf_commute openin_open)
-    then show "openin (subtopology euclidean S) (S \<inter> f -` U)"
+    then show "openin (top_of_set S) (S \<inter> f -` U)"
       by (metis Int_assoc Int_commute assms image_subset_iff_subset_vimage inf.absorb_iff2 vimage_Int)
   qed
 qed
 
 lemma continuous_openin_preimage:
-  "\<lbrakk>continuous_on S f; f ` S \<subseteq> T; openin (subtopology euclidean T) U\<rbrakk>
-        \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U)"
+  "\<lbrakk>continuous_on S f; f ` S \<subseteq> T; openin (top_of_set T) U\<rbrakk>
+        \<Longrightarrow> openin (top_of_set S) (S \<inter> f -` U)"
   by (simp add: continuous_on_open_gen)
 
 lemma continuous_on_closed_gen:
   assumes "f ` S \<subseteq> T"
   shows "continuous_on S f \<longleftrightarrow>
-             (\<forall>U. closedin (subtopology euclidean T) U
-                  \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` U))"
+             (\<forall>U. closedin (top_of_set T) U
+                  \<longrightarrow> closedin (top_of_set S) (S \<inter> f -` U))"
     (is "?lhs = ?rhs")
 proof -
   have *: "U \<subseteq> T \<Longrightarrow> S \<inter> f -` (T - U) = S - (S \<inter> f -` U)" for U
@@ -3868,8 +3957,8 @@
     show ?rhs
     proof clarify
       fix U
-      assume "closedin (subtopology euclidean T) U"
-      then show "closedin (subtopology euclidean S) (S \<inter> f -` U)"
+      assume "closedin (top_of_set T) U"
+      then show "closedin (top_of_set S) (S \<inter> f -` U)"
         using L unfolding continuous_on_open_gen [OF assms]
         by (metis * closedin_def inf_le1 topspace_euclidean_subtopology)
     qed
@@ -3882,16 +3971,376 @@
 qed
 
 lemma continuous_closedin_preimage_gen:
-  assumes "continuous_on S f" "f ` S \<subseteq> T" "closedin (subtopology euclidean T) U"
-    shows "closedin (subtopology euclidean S) (S \<inter> f -` U)"
+  assumes "continuous_on S f" "f ` S \<subseteq> T" "closedin (top_of_set T) U"
+    shows "closedin (top_of_set S) (S \<inter> f -` U)"
 using assms continuous_on_closed_gen by blast
 
 lemma continuous_transform_within_openin:
   assumes "continuous (at a within T) f"
-    and "openin (subtopology euclidean T) S" "a \<in> S"
+    and "openin (top_of_set T) S" "a \<in> S"
     and eq: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
   shows "continuous (at a within T) g"
   using assms by (simp add: Lim_transform_within_openin continuous_within)
 
 
+subsection%important \<open>The topology generated by some (open) subsets\<close>
+
+text \<open>In the definition below of a generated topology, the \<open>Empty\<close> case is not necessary,
+as it follows from \<open>UN\<close> taking for \<open>K\<close> the empty set. However, it is convenient to have,
+and is never a problem in proofs, so I prefer to write it down explicitly.
+
+We do not require \<open>UNIV\<close> to be an open set, as this will not be the case in applications. (We are
+thinking of a topology on a subset of \<open>UNIV\<close>, the remaining part of \<open>UNIV\<close> being irrelevant.)\<close>
+
+inductive generate_topology_on for S where
+  Empty: "generate_topology_on S {}"
+| Int: "generate_topology_on S a \<Longrightarrow> generate_topology_on S b \<Longrightarrow> generate_topology_on S (a \<inter> b)"
+| UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology_on S k) \<Longrightarrow> generate_topology_on S (\<Union>K)"
+| Basis: "s \<in> S \<Longrightarrow> generate_topology_on S s"
+
+lemma istopology_generate_topology_on:
+  "istopology (generate_topology_on S)"
+unfolding istopology_def by (auto intro: generate_topology_on.intros)
+
+text \<open>The basic property of the topology generated by a set \<open>S\<close> is that it is the
+smallest topology containing all the elements of \<open>S\<close>:\<close>
+
+lemma generate_topology_on_coarsest:
+  assumes "istopology T"
+          "\<And>s. s \<in> S \<Longrightarrow> T s"
+          "generate_topology_on S s0"
+  shows "T s0"
+using assms(3) apply (induct rule: generate_topology_on.induct)
+using assms(1) assms(2) unfolding istopology_def by auto
+
+abbreviation%unimportant topology_generated_by::"('a set set) \<Rightarrow> ('a topology)"
+  where "topology_generated_by S \<equiv> topology (generate_topology_on S)"
+
+lemma openin_topology_generated_by_iff:
+  "openin (topology_generated_by S) s \<longleftrightarrow> generate_topology_on S s"
+  using topology_inverse'[OF istopology_generate_topology_on[of S]] by simp
+
+lemma openin_topology_generated_by:
+  "openin (topology_generated_by S) s \<Longrightarrow> generate_topology_on S s"
+using openin_topology_generated_by_iff by auto
+
+lemma topology_generated_by_topspace:
+  "topspace (topology_generated_by S) = (\<Union>S)"
+proof
+  {
+    fix s assume "openin (topology_generated_by S) s"
+    then have "generate_topology_on S s" by (rule openin_topology_generated_by)
+    then have "s \<subseteq> (\<Union>S)" by (induct, auto)
+  }
+  then show "topspace (topology_generated_by S) \<subseteq> (\<Union>S)"
+    unfolding topspace_def by auto
+next
+  have "generate_topology_on S (\<Union>S)"
+    using generate_topology_on.UN[OF generate_topology_on.Basis, of S S] by simp
+  then show "(\<Union>S) \<subseteq> topspace (topology_generated_by S)"
+    unfolding topspace_def using openin_topology_generated_by_iff by auto
+qed
+
+lemma topology_generated_by_Basis:
+  "s \<in> S \<Longrightarrow> openin (topology_generated_by S) s"
+  by (simp only: openin_topology_generated_by_iff, auto simp: generate_topology_on.Basis)
+
+lemma generate_topology_on_Inter:
+  "\<lbrakk>finite \<F>; \<And>K. K \<in> \<F> \<Longrightarrow> generate_topology_on \<S> K; \<F> \<noteq> {}\<rbrakk> \<Longrightarrow> generate_topology_on \<S> (\<Inter>\<F>)"
+  by (induction \<F> rule: finite_induct; force intro: generate_topology_on.intros)
+
+subsection\<open>Topology bases and sub-bases\<close>
+
+lemma istopology_base_alt:
+   "istopology (arbitrary union_of P) \<longleftrightarrow>
+    (\<forall>S T. (arbitrary union_of P) S \<and> (arbitrary union_of P) T
+           \<longrightarrow> (arbitrary union_of P) (S \<inter> T))"
+  by (simp add: istopology_def) (blast intro: arbitrary_union_of_Union)
+
+lemma istopology_base_eq:
+   "istopology (arbitrary union_of P) \<longleftrightarrow>
+    (\<forall>S T. P S \<and> P T \<longrightarrow> (arbitrary union_of P) (S \<inter> T))"
+  by (simp add: istopology_base_alt arbitrary_union_of_Int_eq)
+
+lemma istopology_base:
+   "(\<And>S T. \<lbrakk>P S; P T\<rbrakk> \<Longrightarrow> P(S \<inter> T)) \<Longrightarrow> istopology (arbitrary union_of P)"
+  by (simp add: arbitrary_def istopology_base_eq union_of_inc)
+
+lemma openin_topology_base_unique:
+   "openin X = arbitrary union_of P \<longleftrightarrow>
+        (\<forall>V. P V \<longrightarrow> openin X V) \<and> (\<forall>U x. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V. P V \<and> x \<in> V \<and> V \<subseteq> U))"
+   (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+    by (auto simp: union_of_def arbitrary_def)
+next
+  assume R: ?rhs
+  then have *: "\<exists>\<U>\<subseteq>Collect P. \<Union>\<U> = S" if "openin X S" for S
+    using that by (rule_tac x="{V. P V \<and> V \<subseteq> S}" in exI) fastforce
+  from R show ?lhs
+    by (fastforce simp add: union_of_def arbitrary_def intro: *)
+qed
+
+lemma topology_base_unique:
+   "\<lbrakk>\<And>S. P S \<Longrightarrow> openin X S;
+     \<And>U x. \<lbrakk>openin X U; x \<in> U\<rbrakk> \<Longrightarrow> \<exists>B. P B \<and> x \<in> B \<and> B \<subseteq> U\<rbrakk>
+    \<Longrightarrow> topology(arbitrary union_of P) = X"
+  by (metis openin_topology_base_unique openin_inverse [of X])
+
+lemma topology_bases_eq_aux:
+   "\<lbrakk>(arbitrary union_of P) S;
+     \<And>U x. \<lbrakk>P U; x \<in> U\<rbrakk> \<Longrightarrow> \<exists>V. Q V \<and> x \<in> V \<and> V \<subseteq> U\<rbrakk>
+        \<Longrightarrow> (arbitrary union_of Q) S"
+  by (metis arbitrary_union_of_alt arbitrary_union_of_idempot)
+
+lemma topology_bases_eq:
+   "\<lbrakk>\<And>U x. \<lbrakk>P U; x \<in> U\<rbrakk> \<Longrightarrow> \<exists>V. Q V \<and> x \<in> V \<and> V \<subseteq> U;
+    \<And>V x. \<lbrakk>Q V; x \<in> V\<rbrakk> \<Longrightarrow> \<exists>U. P U \<and> x \<in> U \<and> U \<subseteq> V\<rbrakk>
+        \<Longrightarrow> topology (arbitrary union_of P) =
+            topology (arbitrary union_of Q)"
+  by (fastforce intro:  arg_cong [where f=topology]  elim: topology_bases_eq_aux)
+
+lemma istopology_subbase:
+   "istopology (arbitrary union_of (finite intersection_of P relative_to S))"
+  by (simp add: finite_intersection_of_Int istopology_base relative_to_Int)
+
+lemma openin_subbase:
+  "openin (topology (arbitrary union_of (finite intersection_of B relative_to U))) S
+   \<longleftrightarrow> (arbitrary union_of (finite intersection_of B relative_to U)) S"
+  by (simp add: istopology_subbase topology_inverse')
+
+lemma topspace_subbase [simp]:
+   "topspace(topology (arbitrary union_of (finite intersection_of B relative_to U))) = U" (is "?lhs = _")
+proof
+  show "?lhs \<subseteq> U"
+    by (metis arbitrary_union_of_relative_to openin_subbase openin_topspace relative_to_imp_subset)
+  show "U \<subseteq> ?lhs"
+    by (metis arbitrary_union_of_inc finite_intersection_of_empty inf.orderE istopology_subbase 
+              openin_subset relative_to_inc subset_UNIV topology_inverse')
+qed
+
+lemma minimal_topology_subbase:
+   "\<lbrakk>\<And>S. P S \<Longrightarrow> openin X S; openin X U;
+     openin(topology(arbitrary union_of (finite intersection_of P relative_to U))) S\<rbrakk>
+    \<Longrightarrow> openin X S"
+  apply (simp add: istopology_subbase topology_inverse)
+  apply (simp add: union_of_def intersection_of_def relative_to_def)
+  apply (blast intro: openin_Int_Inter)
+  done
+
+lemma istopology_subbase_UNIV:
+   "istopology (arbitrary union_of (finite intersection_of P))"
+  by (simp add: istopology_base finite_intersection_of_Int)
+
+
+lemma generate_topology_on_eq:
+  "generate_topology_on S = arbitrary union_of finite' intersection_of (\<lambda>x. x \<in> S)" (is "?lhs = ?rhs")
+proof (intro ext iffI)
+  fix A
+  assume "?lhs A"
+  then show "?rhs A"
+  proof induction
+    case (Int a b)
+    then show ?case
+      by (metis (mono_tags, lifting) istopology_base_alt finite'_intersection_of_Int istopology_base)
+  next
+    case (UN K)
+    then show ?case
+      by (simp add: arbitrary_union_of_Union)
+  next
+    case (Basis s)
+    then show ?case
+      by (simp add: Sup_upper arbitrary_union_of_inc finite'_intersection_of_inc relative_to_subset)
+  qed auto
+next
+  fix A
+  assume "?rhs A"
+  then obtain \<U> where \<U>: "\<And>T. T \<in> \<U> \<Longrightarrow> \<exists>\<F>. finite' \<F> \<and> \<F> \<subseteq> S \<and> \<Inter>\<F> = T" and eq: "A = \<Union>\<U>"
+    unfolding union_of_def intersection_of_def by auto
+  show "?lhs A"
+    unfolding eq
+  proof (rule generate_topology_on.UN)
+    fix T
+    assume "T \<in> \<U>"
+    with \<U> obtain \<F> where "finite' \<F>" "\<F> \<subseteq> S" "\<Inter>\<F> = T"
+      by blast
+    have "generate_topology_on S (\<Inter>\<F>)"
+    proof (rule generate_topology_on_Inter)
+      show "finite \<F>" "\<F> \<noteq> {}"
+        by (auto simp: \<open>finite' \<F>\<close>)
+      show "\<And>K. K \<in> \<F> \<Longrightarrow> generate_topology_on S K"
+        by (metis \<open>\<F> \<subseteq> S\<close> generate_topology_on.simps subset_iff)
+    qed
+    then show "generate_topology_on S T"
+      using \<open>\<Inter>\<F> = T\<close> by blast
+  qed
+qed
+
+subsubsection%important \<open>Continuity\<close>
+
+text \<open>We will need to deal with continuous maps in terms of topologies and not in terms
+of type classes, as defined below.\<close>
+
+definition%important continuous_on_topo::"'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
+  where "continuous_on_topo T1 T2 f = ((\<forall> U. openin T2 U \<longrightarrow> openin T1 (f-`U \<inter> topspace(T1)))
+                                      \<and> (f`(topspace T1) \<subseteq> (topspace T2)))"
+
+lemma continuous_on_continuous_on_topo:
+  "continuous_on s f \<longleftrightarrow> continuous_on_topo (top_of_set s) euclidean f"
+  by (auto simp: continuous_on_topo_def Int_commute continuous_openin_preimage_eq)
+
+lemma continuous_on_topo_UNIV:
+  "continuous_on UNIV f \<longleftrightarrow> continuous_on_topo euclidean euclidean f"
+using continuous_on_continuous_on_topo[of UNIV f] subtopology_UNIV[of euclidean] by auto
+
+lemma continuous_on_topo_open [intro]:
+  "continuous_on_topo T1 T2 f \<Longrightarrow> openin T2 U \<Longrightarrow> openin T1 (f-`U \<inter> topspace(T1))"
+  unfolding continuous_on_topo_def by auto
+
+lemma continuous_on_topo_topspace [intro]:
+  "continuous_on_topo T1 T2 f \<Longrightarrow> f`(topspace T1) \<subseteq> (topspace T2)"
+unfolding continuous_on_topo_def by auto
+
+lemma continuous_on_generated_topo_iff:
+  "continuous_on_topo T1 (topology_generated_by S) f \<longleftrightarrow>
+      ((\<forall>U. U \<in> S \<longrightarrow> openin T1 (f-`U \<inter> topspace(T1))) \<and> (f`(topspace T1) \<subseteq> (\<Union> S)))"
+unfolding continuous_on_topo_def topology_generated_by_topspace
+proof (auto simp add: topology_generated_by_Basis)
+  assume H: "\<forall>U. U \<in> S \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)"
+  fix U assume "openin (topology_generated_by S) U"
+  then have "generate_topology_on S U" by (rule openin_topology_generated_by)
+  then show "openin T1 (f -` U \<inter> topspace T1)"
+  proof (induct)
+    fix a b
+    assume H: "openin T1 (f -` a \<inter> topspace T1)" "openin T1 (f -` b \<inter> topspace T1)"
+    have "f -` (a \<inter> b) \<inter> topspace T1 = (f-`a \<inter> topspace T1) \<inter> (f-`b \<inter> topspace T1)"
+      by auto
+    then show "openin T1 (f -` (a \<inter> b) \<inter> topspace T1)" using H by auto
+  next
+    fix K
+    assume H: "openin T1 (f -` k \<inter> topspace T1)" if "k\<in> K" for k
+    define L where "L = {f -` k \<inter> topspace T1|k. k \<in> K}"
+    have *: "openin T1 l" if "l \<in>L" for l using that H unfolding L_def by auto
+    have "openin T1 (\<Union>L)" using openin_Union[OF *] by simp
+    moreover have "(\<Union>L) = (f -` \<Union>K \<inter> topspace T1)" unfolding L_def by auto
+    ultimately show "openin T1 (f -` \<Union>K \<inter> topspace T1)" by simp
+  qed (auto simp add: H)
+qed
+
+lemma continuous_on_generated_topo:
+  assumes "\<And>U. U \<in>S \<Longrightarrow> openin T1 (f-`U \<inter> topspace(T1))"
+          "f`(topspace T1) \<subseteq> (\<Union> S)"
+  shows "continuous_on_topo T1 (topology_generated_by S) f"
+  using assms continuous_on_generated_topo_iff by blast
+
+proposition continuous_on_topo_compose:
+  assumes "continuous_on_topo T1 T2 f" "continuous_on_topo T2 T3 g"
+  shows "continuous_on_topo T1 T3 (g o f)"
+  using assms unfolding continuous_on_topo_def
+proof (auto)
+  fix U :: "'c set"
+  assume H: "openin T3 U"
+  have "openin T1 (f -` (g -` U \<inter> topspace T2) \<inter> topspace T1)"
+    using H assms by blast
+  moreover have "f -` (g -` U \<inter> topspace T2) \<inter> topspace T1 = (g \<circ> f) -` U \<inter> topspace T1"
+    using H assms continuous_on_topo_topspace by fastforce
+  ultimately show "openin T1 ((g \<circ> f) -` U \<inter> topspace T1)"
+    by simp
+qed (blast)
+
+lemma continuous_on_topo_preimage_topspace [intro]:
+  assumes "continuous_on_topo T1 T2 f"
+  shows "f-`(topspace T2) \<inter> topspace T1 = topspace T1"
+using assms unfolding continuous_on_topo_def by auto
+
+
+subsubsection%important \<open>Pullback topology\<close>
+
+text \<open>Pulling back a topology by map gives again a topology. \<open>subtopology\<close> is
+a special case of this notion, pulling back by the identity. We introduce the general notion as
+we will need it to define the strong operator topology on the space of continuous linear operators,
+by pulling back the product topology on the space of all functions.\<close>
+
+text \<open>\<open>pullback_topology A f T\<close> is the pullback of the topology \<open>T\<close> by the map \<open>f\<close> on
+the set \<open>A\<close>.\<close>
+
+definition%important pullback_topology::"('a set) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b topology) \<Rightarrow> ('a topology)"
+  where "pullback_topology A f T = topology (\<lambda>S. \<exists>U. openin T U \<and> S = f-`U \<inter> A)"
+
+lemma istopology_pullback_topology:
+  "istopology (\<lambda>S. \<exists>U. openin T U \<and> S = f-`U \<inter> A)"
+  unfolding istopology_def proof (auto)
+  fix K assume "\<forall>S\<in>K. \<exists>U. openin T U \<and> S = f -` U \<inter> A"
+  then have "\<exists>U. \<forall>S\<in>K. openin T (U S) \<and> S = f-`(U S) \<inter> A"
+    by (rule bchoice)
+  then obtain U where U: "\<forall>S\<in>K. openin T (U S) \<and> S = f-`(U S) \<inter> A"
+    by blast
+  define V where "V = (\<Union>S\<in>K. U S)"
+  have "openin T V" "\<Union>K = f -` V \<inter> A" unfolding V_def using U by auto
+  then show "\<exists>V. openin T V \<and> \<Union>K = f -` V \<inter> A" by auto
+qed
+
+lemma openin_pullback_topology:
+  "openin (pullback_topology A f T) S \<longleftrightarrow> (\<exists>U. openin T U \<and> S = f-`U \<inter> A)"
+unfolding pullback_topology_def topology_inverse'[OF istopology_pullback_topology] by auto
+
+lemma topspace_pullback_topology:
+  "topspace (pullback_topology A f T) = f-`(topspace T) \<inter> A"
+by (auto simp add: topspace_def openin_pullback_topology)
+
+proposition continuous_on_topo_pullback [intro]:
+  assumes "continuous_on_topo T1 T2 g"
+  shows "continuous_on_topo (pullback_topology A f T1) T2 (g o f)"
+unfolding continuous_on_topo_def
+proof (auto)
+  fix U::"'b set" assume "openin T2 U"
+  then have "openin T1 (g-`U \<inter> topspace T1)"
+    using assms unfolding continuous_on_topo_def by auto
+  have "(g o f)-`U \<inter> topspace (pullback_topology A f T1) = (g o f)-`U \<inter> A \<inter> f-`(topspace T1)"
+    unfolding topspace_pullback_topology by auto
+  also have "... = f-`(g-`U \<inter> topspace T1) \<inter> A "
+    by auto
+  also have "openin (pullback_topology A f T1) (...)"
+    unfolding openin_pullback_topology using \<open>openin T1 (g-`U \<inter> topspace T1)\<close> by auto
+  finally show "openin (pullback_topology A f T1) ((g \<circ> f) -` U \<inter> topspace (pullback_topology A f T1))"
+    by auto
+next
+  fix x assume "x \<in> topspace (pullback_topology A f T1)"
+  then have "f x \<in> topspace T1"
+    unfolding topspace_pullback_topology by auto
+  then show "g (f x) \<in> topspace T2"
+    using assms unfolding continuous_on_topo_def by auto
+qed
+
+proposition continuous_on_topo_pullback' [intro]:
+  assumes "continuous_on_topo T1 T2 (f o g)" "topspace T1 \<subseteq> g-`A"
+  shows "continuous_on_topo T1 (pullback_topology A f T2) g"
+unfolding continuous_on_topo_def
+proof (auto)
+  fix U assume "openin (pullback_topology A f T2) U"
+  then have "\<exists>V. openin T2 V \<and> U = f-`V \<inter> A"
+    unfolding openin_pullback_topology by auto
+  then obtain V where "openin T2 V" "U = f-`V \<inter> A"
+    by blast
+  then have "g -` U \<inter> topspace T1 = g-`(f-`V \<inter> A) \<inter> topspace T1"
+    by blast
+  also have "... = (f o g)-`V \<inter> (g-`A \<inter> topspace T1)"
+    by auto
+  also have "... = (f o g)-`V \<inter> topspace T1"
+    using assms(2) by auto
+  also have "openin T1 (...)"
+    using assms(1) \<open>openin T2 V\<close> by auto
+  finally show "openin T1 (g -` U \<inter> topspace T1)" by simp
+next
+  fix x assume "x \<in> topspace T1"
+  have "(f o g) x \<in> topspace T2"
+    using assms(1) \<open>x \<in> topspace T1\<close> unfolding continuous_on_topo_def by auto
+  then have "g x \<in> f-`(topspace T2)"
+    unfolding comp_def by blast
+  moreover have "g x \<in> A" using assms(2) \<open>x \<in> topspace T1\<close> by blast
+  ultimately show "g x \<in> topspace (pullback_topology A f T2)"
+    unfolding topspace_pullback_topology by blast
+qed
+
 end
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Abstract_Topology_2.thy
--- a/src/HOL/Analysis/Abstract_Topology_2.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Abstract_Topology_2.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -99,14 +99,14 @@
 qed
 
 lemma closedin_limpt:
-  "closedin (subtopology euclidean T) S \<longleftrightarrow> S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S)"
+  "closedin (top_of_set T) S \<longleftrightarrow> S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S)"
   apply (simp add: closedin_closed, safe)
    apply (simp add: closed_limpt islimpt_subset)
   apply (rule_tac x="closure S" in exI, simp)
   apply (force simp: closure_def)
   done
 
-lemma closedin_closed_eq: "closed S \<Longrightarrow> closedin (subtopology euclidean S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S"
+lemma closedin_closed_eq: "closed S \<Longrightarrow> closedin (top_of_set S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S"
   by (meson closedin_limpt closed_subset closedin_closed_trans)
 
 lemma connected_closed_set:
@@ -123,32 +123,35 @@
 by (metis assms closed_Un connected_closed_set)
 
 lemma closedin_subset_trans:
-  "closedin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
-    closedin (subtopology euclidean T) S"
+  "closedin (top_of_set U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
+    closedin (top_of_set T) S"
   by (meson closedin_limpt subset_iff)
 
 lemma openin_subset_trans:
-  "openin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
-    openin (subtopology euclidean T) S"
+  "openin (top_of_set U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
+    openin (top_of_set T) S"
   by (auto simp: openin_open)
 
 lemma closedin_compact:
-   "\<lbrakk>compact S; closedin (subtopology euclidean S) T\<rbrakk> \<Longrightarrow> compact T"
+   "\<lbrakk>compact S; closedin (top_of_set S) T\<rbrakk> \<Longrightarrow> compact T"
 by (metis closedin_closed compact_Int_closed)
 
 lemma closedin_compact_eq:
   fixes S :: "'a::t2_space set"
   shows
    "compact S
-         \<Longrightarrow> (closedin (subtopology euclidean S) T \<longleftrightarrow>
+         \<Longrightarrow> (closedin (top_of_set S) T \<longleftrightarrow>
               compact T \<and> T \<subseteq> S)"
 by (metis closedin_imp_subset closedin_compact closed_subset compact_imp_closed)
 
 
 subsection \<open>Closure\<close>
 
+lemma euclidean_closure_of [simp]: "euclidean closure_of S = closure S"
+  by (auto simp: closure_of_def closure_def islimpt_def)
+
 lemma closure_openin_Int_closure:
-  assumes ope: "openin (subtopology euclidean U) S" and "T \<subseteq> U"
+  assumes ope: "openin (top_of_set U) S" and "T \<subseteq> U"
   shows "closure(S \<inter> closure T) = closure(S \<inter> T)"
 proof
   obtain V where "open V" and S: "S = U \<inter> V"
@@ -171,16 +174,16 @@
 
 corollary infinite_openin:
   fixes S :: "'a :: t1_space set"
-  shows "\<lbrakk>openin (subtopology euclidean U) S; x \<in> S; x islimpt U\<rbrakk> \<Longrightarrow> infinite S"
+  shows "\<lbrakk>openin (top_of_set U) S; x \<in> S; x islimpt U\<rbrakk> \<Longrightarrow> infinite S"
   by (clarsimp simp add: openin_open islimpt_eq_acc_point inf_commute)
 
 lemma closure_Int_ballI:
-  assumes "\<And>U. \<lbrakk>openin (subtopology euclidean S) U; U \<noteq> {}\<rbrakk> \<Longrightarrow> T \<inter> U \<noteq> {}"
+  assumes "\<And>U. \<lbrakk>openin (top_of_set S) U; U \<noteq> {}\<rbrakk> \<Longrightarrow> T \<inter> U \<noteq> {}"
   shows "S \<subseteq> closure T"
 proof (clarsimp simp: closure_iff_nhds_not_empty)
   fix x and A and V
   assume "x \<in> S" "V \<subseteq> A" "open V" "x \<in> V" "T \<inter> A = {}"
-  then have "openin (subtopology euclidean S) (A \<inter> V \<inter> S)"
+  then have "openin (top_of_set S) (A \<inter> V \<inter> S)"
     by (auto simp: openin_open intro!: exI[where x="V"])
   moreover have "A \<inter> V \<inter> S \<noteq> {}" using \<open>x \<in> V\<close> \<open>V \<subseteq> A\<close> \<open>x \<in> S\<close>
     by auto
@@ -192,6 +195,12 @@
 
 subsection \<open>Frontier\<close>
 
+lemma euclidean_interior_of [simp]: "euclidean interior_of S = interior S"
+  by (auto simp: interior_of_def interior_def)
+
+lemma euclidean_frontier_of [simp]: "euclidean frontier_of S = frontier S"
+  by (auto simp: frontier_of_def frontier_def)
+
 lemma connected_Int_frontier:
      "\<lbrakk>connected s; s \<inter> t \<noteq> {}; s - t \<noteq> {}\<rbrakk> \<Longrightarrow> (s \<inter> frontier t \<noteq> {})"
   apply (simp add: frontier_interiors connected_openin, safe)
@@ -204,17 +213,17 @@
 
 lemma openin_delete:
   fixes a :: "'a :: t1_space"
-  shows "openin (subtopology euclidean u) s
-         \<Longrightarrow> openin (subtopology euclidean u) (s - {a})"
+  shows "openin (top_of_set u) s
+         \<Longrightarrow> openin (top_of_set u) (s - {a})"
 by (metis Int_Diff open_delete openin_open)
 
 lemma compact_eq_openin_cover:
   "compact S \<longleftrightarrow>
-    (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
+    (\<forall>C. (\<forall>c\<in>C. openin (top_of_set S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
 proof safe
   fix C
-  assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"
+  assume "compact S" and "\<forall>c\<in>C. openin (top_of_set S) c" and "S \<subseteq> \<Union>C"
   then have "\<forall>c\<in>{T. open T \<and> S \<inter> T \<in> C}. open c" and "S \<subseteq> \<Union>{T. open T \<and> S \<inter> T \<in> C}"
     unfolding openin_open by force+
   with \<open>compact S\<close> obtain D where "D \<subseteq> {T. open T \<and> S \<inter> T \<in> C}" and "finite D" and "S \<subseteq> \<Union>D"
@@ -223,14 +232,14 @@
     by auto
   then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
 next
-  assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
+  assume 1: "\<forall>C. (\<forall>c\<in>C. openin (top_of_set S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"
   show "compact S"
   proof (rule compactI)
     fix C
     let ?C = "image (\<lambda>T. S \<inter> T) C"
     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"
-    then have "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"
+    then have "(\<forall>c\<in>?C. openin (top_of_set S) c) \<and> S \<subseteq> \<Union>?C"
       unfolding openin_open by auto
     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"
       by metis
@@ -276,7 +285,7 @@
 
 lemma continuous_closedin_preimage_constant:
   fixes f :: "_ \<Rightarrow> 'b::t1_space"
-  shows "continuous_on S f \<Longrightarrow> closedin (subtopology euclidean S) {x \<in> S. f x = a}"
+  shows "continuous_on S f \<Longrightarrow> closedin (top_of_set S) {x \<in> S. f x = a}"
   using continuous_closedin_preimage[of S f "{a}"] by (simp add: vimage_def Collect_conj_eq)
 
 lemma continuous_closed_preimage_constant:
@@ -335,14 +344,14 @@
 subsection%unimportant \<open>Continuity relative to a union.\<close>
 
 lemma continuous_on_Un_local:
-    "\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;
+    "\<lbrakk>closedin (top_of_set (s \<union> t)) s; closedin (top_of_set (s \<union> t)) t;
       continuous_on s f; continuous_on t f\<rbrakk>
      \<Longrightarrow> continuous_on (s \<union> t) f"
   unfolding continuous_on closedin_limpt
   by (metis Lim_trivial_limit Lim_within_union Un_iff trivial_limit_within)
 
 lemma continuous_on_cases_local:
-     "\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;
+     "\<lbrakk>closedin (top_of_set (s \<union> t)) s; closedin (top_of_set (s \<union> t)) t;
        continuous_on s f; continuous_on t g;
        \<And>x. \<lbrakk>x \<in> s \<and> \<not>P x \<or> x \<in> t \<and> P x\<rbrakk> \<Longrightarrow> f x = g x\<rbrakk>
       \<Longrightarrow> continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
@@ -358,9 +367,9 @@
 proof -
   have s: "s = (s \<inter> h -` atMost a) \<union> (s \<inter> h -` atLeast a)"
     by force
-  have 1: "closedin (subtopology euclidean s) (s \<inter> h -` atMost a)"
+  have 1: "closedin (top_of_set s) (s \<inter> h -` atMost a)"
     by (rule continuous_closedin_preimage [OF h closed_atMost])
-  have 2: "closedin (subtopology euclidean s) (s \<inter> h -` atLeast a)"
+  have 2: "closedin (top_of_set s) (s \<inter> h -` atLeast a)"
     by (rule continuous_closedin_preimage [OF h closed_atLeast])
   have eq: "s \<inter> h -` {..a} = {t \<in> s. h t \<le> a}" "s \<inter> h -` {a..} = {t \<in> s. a \<le> h t}"
     by auto
@@ -388,7 +397,7 @@
   assumes contf: "continuous_on S f"
     and imf: "f ` S = T"
     and injf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
-    and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
+    and oo: "\<And>U. openin (top_of_set S) U \<Longrightarrow> openin (top_of_set T) (f ` U)"
   shows "continuous_on T g"
 proof -
   from imf injf have gTS: "g ` T = S"
@@ -403,7 +412,7 @@
   assumes contf: "continuous_on S f"
     and imf: "f ` S = T"
     and injf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
-    and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
+    and oo: "\<And>U. closedin (top_of_set S) U \<Longrightarrow> closedin (top_of_set T) (f ` U)"
   shows "continuous_on T g"
 proof -
   from imf injf have gTS: "g ` T = S"
@@ -418,7 +427,7 @@
   assumes contf: "continuous_on S f"
     and imf: "f ` S = T"
     and injf: "inj_on f S"
-    and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
+    and oo: "\<And>U. openin (top_of_set S) U \<Longrightarrow> openin (top_of_set T) (f ` U)"
   obtains g where "homeomorphism S T f g"
 proof
   have "continuous_on T (inv_into S f)"
@@ -431,7 +440,7 @@
   assumes contf: "continuous_on S f"
     and imf: "f ` S = T"
     and injf: "inj_on f S"
-    and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
+    and oo: "\<And>U. closedin (top_of_set S) U \<Longrightarrow> closedin (top_of_set T) (f ` U)"
   obtains g where "homeomorphism S T f g"
 proof
   have "continuous_on T (inv_into S f)"
@@ -442,8 +451,8 @@
 
 lemma homeomorphism_imp_open_map:
   assumes hom: "homeomorphism S T f g"
-    and oo: "openin (subtopology euclidean S) U"
-  shows "openin (subtopology euclidean T) (f ` U)"
+    and oo: "openin (top_of_set S) U"
+  shows "openin (top_of_set T) (f ` U)"
 proof -
   from hom oo have [simp]: "f ` U = T \<inter> g -` U"
     using openin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
@@ -457,8 +466,8 @@
 
 lemma homeomorphism_imp_closed_map:
   assumes hom: "homeomorphism S T f g"
-    and oo: "closedin (subtopology euclidean S) U"
-  shows "closedin (subtopology euclidean T) (f ` U)"
+    and oo: "closedin (top_of_set S) U"
+  shows "closedin (top_of_set T) (f ` U)"
 proof -
   from hom oo have [simp]: "f ` U = T \<inter> g -` U"
     using closedin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
@@ -476,13 +485,13 @@
   obtains \<B> :: "'a:: second_countable_topology set set"
     where "countable \<B>"
           "{} \<notin> \<B>"
-          "\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"
-          "\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
+          "\<And>C. C \<in> \<B> \<Longrightarrow> openin(top_of_set S) C"
+          "\<And>T. openin(top_of_set S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
 proof -
   obtain \<B> :: "'a set set"
     where "countable \<B>"
-      and opeB: "\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"
-      and \<B>:    "\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
+      and opeB: "\<And>C. C \<in> \<B> \<Longrightarrow> openin(top_of_set S) C"
+      and \<B>:    "\<And>T. openin(top_of_set S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
   proof -
     obtain \<C> :: "'a set set"
       where "countable \<C>" and ope: "\<And>C. C \<in> \<C> \<Longrightarrow> open C"
@@ -492,9 +501,9 @@
     proof
       show "countable ((\<lambda>C. S \<inter> C) ` \<C>)"
         by (simp add: \<open>countable \<C>\<close>)
-      show "\<And>C. C \<in> (\<inter>) S ` \<C> \<Longrightarrow> openin (subtopology euclidean S) C"
+      show "\<And>C. C \<in> (\<inter>) S ` \<C> \<Longrightarrow> openin (top_of_set S) C"
         using ope by auto
-      show "\<And>T. openin (subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>\<subseteq>(\<inter>) S ` \<C>. T = \<Union>\<U>"
+      show "\<And>T. openin (top_of_set S) T \<Longrightarrow> \<exists>\<U>\<subseteq>(\<inter>) S ` \<C>. T = \<Union>\<U>"
         by (metis \<C> image_mono inf_Sup openin_open)
     qed
   qed
@@ -502,9 +511,9 @@
   proof
     show "countable (\<B> - {{}})"
       using \<open>countable \<B>\<close> by blast
-    show "\<And>C. \<lbrakk>C \<in> \<B> - {{}}\<rbrakk> \<Longrightarrow> openin (subtopology euclidean S) C"
-      by (simp add: \<open>\<And>C. C \<in> \<B> \<Longrightarrow> openin (subtopology euclidean S) C\<close>)
-    show "\<exists>\<U>\<subseteq>\<B> - {{}}. T = \<Union>\<U>" if "openin (subtopology euclidean S) T" for T
+    show "\<And>C. \<lbrakk>C \<in> \<B> - {{}}\<rbrakk> \<Longrightarrow> openin (top_of_set S) C"
+      by (simp add: \<open>\<And>C. C \<in> \<B> \<Longrightarrow> openin (top_of_set S) C\<close>)
+    show "\<exists>\<U>\<subseteq>\<B> - {{}}. T = \<Union>\<U>" if "openin (top_of_set S) T" for T
       using \<B> [OF that]
       apply clarify
       apply (rule_tac x="\<U> - {{}}" in exI, auto)
@@ -514,7 +523,7 @@
 
 lemma Lindelof_openin:
   fixes \<F> :: "'a::second_countable_topology set set"
-  assumes "\<And>S. S \<in> \<F> \<Longrightarrow> openin (subtopology euclidean U) S"
+  assumes "\<And>S. S \<in> \<F> \<Longrightarrow> openin (top_of_set U) S"
   obtains \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
 proof -
   have "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>T. open T \<and> S = U \<inter> T"
@@ -535,16 +544,16 @@
 
 lemma continuous_imp_closed_map:
   fixes f :: "'a::t2_space \<Rightarrow> 'b::t2_space"
-  assumes "closedin (subtopology euclidean S) U"
+  assumes "closedin (top_of_set S) U"
           "continuous_on S f" "f ` S = T" "compact S"
-    shows "closedin (subtopology euclidean T) (f ` U)"
+    shows "closedin (top_of_set T) (f ` U)"
   by (metis assms closedin_compact_eq compact_continuous_image continuous_on_subset subset_image_iff)
 
 lemma closed_map_restrict:
-  assumes cloU: "closedin (subtopology euclidean (S \<inter> f -` T')) U"
-    and cc: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
+  assumes cloU: "closedin (top_of_set (S \<inter> f -` T')) U"
+    and cc: "\<And>U. closedin (top_of_set S) U \<Longrightarrow> closedin (top_of_set T) (f ` U)"
     and "T' \<subseteq> T"
-  shows "closedin (subtopology euclidean T') (f ` U)"
+  shows "closedin (top_of_set T') (f ` U)"
 proof -
   obtain V where "closed V" "U = S \<inter> f -` T' \<inter> V"
     using cloU by (auto simp: closedin_closed)
@@ -555,10 +564,10 @@
 subsection%unimportant\<open>Open Maps\<close>
 
 lemma open_map_restrict:
-  assumes opeU: "openin (subtopology euclidean (S \<inter> f -` T')) U"
-    and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
+  assumes opeU: "openin (top_of_set (S \<inter> f -` T')) U"
+    and oo: "\<And>U. openin (top_of_set S) U \<Longrightarrow> openin (top_of_set T) (f ` U)"
     and "T' \<subseteq> T"
-  shows "openin (subtopology euclidean T') (f ` U)"
+  shows "openin (top_of_set T') (f ` U)"
 proof -
   obtain V where "open V" "U = S \<inter> f -` T' \<inter> V"
     using opeU by (auto simp: openin_open)
@@ -572,8 +581,8 @@
 lemma quotient_map_imp_continuous_open:
   assumes T: "f ` S \<subseteq> T"
       and ope: "\<And>U. U \<subseteq> T
-              \<Longrightarrow> (openin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
-                   openin (subtopology euclidean T) U)"
+              \<Longrightarrow> (openin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
+                   openin (top_of_set T) U)"
     shows "continuous_on S f"
 proof -
   have [simp]: "S \<inter> f -` f ` S = S" by auto
@@ -586,8 +595,8 @@
 lemma quotient_map_imp_continuous_closed:
   assumes T: "f ` S \<subseteq> T"
       and ope: "\<And>U. U \<subseteq> T
-                  \<Longrightarrow> (closedin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
-                       closedin (subtopology euclidean T) U)"
+                  \<Longrightarrow> (closedin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
+                       closedin (top_of_set T) U)"
     shows "continuous_on S f"
 proof -
   have [simp]: "S \<inter> f -` f ` S = S" by auto
@@ -600,10 +609,10 @@
 lemma open_map_imp_quotient_map:
   assumes contf: "continuous_on S f"
       and T: "T \<subseteq> f ` S"
-      and ope: "\<And>T. openin (subtopology euclidean S) T
-                   \<Longrightarrow> openin (subtopology euclidean (f ` S)) (f ` T)"
-    shows "openin (subtopology euclidean S) (S \<inter> f -` T) =
-           openin (subtopology euclidean (f ` S)) T"
+      and ope: "\<And>T. openin (top_of_set S) T
+                   \<Longrightarrow> openin (top_of_set (f ` S)) (f ` T)"
+    shows "openin (top_of_set S) (S \<inter> f -` T) =
+           openin (top_of_set (f ` S)) T"
 proof -
   have "T = f ` (S \<inter> f -` T)"
     using T by blast
@@ -614,14 +623,14 @@
 lemma closed_map_imp_quotient_map:
   assumes contf: "continuous_on S f"
       and T: "T \<subseteq> f ` S"
-      and ope: "\<And>T. closedin (subtopology euclidean S) T
-              \<Longrightarrow> closedin (subtopology euclidean (f ` S)) (f ` T)"
-    shows "openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow>
-           openin (subtopology euclidean (f ` S)) T"
+      and ope: "\<And>T. closedin (top_of_set S) T
+              \<Longrightarrow> closedin (top_of_set (f ` S)) (f ` T)"
+    shows "openin (top_of_set S) (S \<inter> f -` T) \<longleftrightarrow>
+           openin (top_of_set (f ` S)) T"
           (is "?lhs = ?rhs")
 proof
   assume ?lhs
-  then have *: "closedin (subtopology euclidean S) (S - (S \<inter> f -` T))"
+  then have *: "closedin (top_of_set S) (S - (S \<inter> f -` T))"
     using closedin_diff by fastforce
   have [simp]: "(f ` S - f ` (S - (S \<inter> f -` T))) = T"
     using T by blast
@@ -638,14 +647,14 @@
       and contg: "continuous_on T g" and img: "g ` T \<subseteq> S"
       and fg [simp]: "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"
       and U: "U \<subseteq> T"
-    shows "openin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
-           openin (subtopology euclidean T) U"
+    shows "openin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
+           openin (top_of_set T) U"
           (is "?lhs = ?rhs")
 proof -
-  have f: "\<And>Z. openin (subtopology euclidean (f ` S)) Z \<Longrightarrow>
-                openin (subtopology euclidean S) (S \<inter> f -` Z)"
-  and  g: "\<And>Z. openin (subtopology euclidean (g ` T)) Z \<Longrightarrow>
-                openin (subtopology euclidean T) (T \<inter> g -` Z)"
+  have f: "\<And>Z. openin (top_of_set (f ` S)) Z \<Longrightarrow>
+                openin (top_of_set S) (S \<inter> f -` Z)"
+  and  g: "\<And>Z. openin (top_of_set (g ` T)) Z \<Longrightarrow>
+                openin (top_of_set T) (T \<inter> g -` Z)"
     using contf contg by (auto simp: continuous_on_open)
   show ?thesis
   proof
@@ -655,7 +664,7 @@
       using U by auto
     finally have eq: "T \<inter> g -` (g ` T \<inter> (S \<inter> f -` U)) = U" .
     assume ?lhs
-    then have *: "openin (subtopology euclidean (g ` T)) (g ` T \<inter> (S \<inter> f -` U))"
+    then have *: "openin (top_of_set (g ` T)) (g ` T \<inter> (S \<inter> f -` U))"
       by (meson img openin_Int openin_subtopology_Int_subset openin_subtopology_self)
     show ?rhs
       using g [OF *] eq by auto
@@ -671,8 +680,8 @@
       and "continuous_on (f ` S) g"
       and  "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
       and "U \<subseteq> f ` S"
-    shows "openin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
-           openin (subtopology euclidean (f ` S)) U"
+    shows "openin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
+           openin (top_of_set (f ` S)) U"
 apply (rule continuous_right_inverse_imp_quotient_map)
 using assms apply force+
 done
@@ -680,101 +689,220 @@
 lemma continuous_imp_quotient_map:
   fixes f :: "'a::t2_space \<Rightarrow> 'b::t2_space"
   assumes "continuous_on S f" "f ` S = T" "compact S" "U \<subseteq> T"
-    shows "openin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
-           openin (subtopology euclidean T) U"
+    shows "openin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
+           openin (top_of_set T) U"
   by (metis (no_types, lifting) assms closed_map_imp_quotient_map continuous_imp_closed_map)
 
-subsection%unimportant\<open>Pasting functions together\<close>
+subsection%unimportant\<open>Pasting lemmas for functions, for of casewise definitions\<close>
 
-text\<open>on open sets\<close>
+subsubsection\<open>on open sets\<close>
 
 lemma pasting_lemma:
-  fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
-  assumes clo: "\<And>i. i \<in> I \<Longrightarrow> openin (subtopology euclidean S) (T i)"
-      and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
-      and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
-      and g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
-    shows "continuous_on S g"
-proof (clarsimp simp: continuous_openin_preimage_eq)
-  fix U :: "'b set"
-  assume "open U"
-  have S: "\<And>i. i \<in> I \<Longrightarrow> (T i) \<subseteq> S"
-    using clo openin_imp_subset by blast
-  have *: "(S \<inter> g -` U) = (\<Union>i \<in> I. T i \<inter> f i -` U)"
-    using S f g by fastforce
-  show "openin (subtopology euclidean S) (S \<inter> g -` U)"
-    apply (subst *)
-    apply (rule openin_Union, clarify)
-    using \<open>open U\<close> clo cont continuous_openin_preimage_gen openin_trans by blast
+  assumes ope: "\<And>i. i \<in> I \<Longrightarrow> openin X (T i)"
+      and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_map(subtopology X (T i)) Y (f i)"
+      and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
+      and g: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
+    shows "continuous_map X Y g"
+  unfolding continuous_map_openin_preimage_eq
+proof (intro conjI allI impI)
+  show "g ` topspace X \<subseteq> topspace Y"
+    using g cont continuous_map_image_subset_topspace topspace_subtopology by fastforce
+next
+  fix U
+  assume Y: "openin Y U"
+  have T: "T i \<subseteq> topspace X" if "i \<in> I" for i
+    using ope by (simp add: openin_subset that)
+  have *: "topspace X \<inter> g -` U = (\<Union>i \<in> I. T i \<inter> f i -` U)"
+    using f g T by fastforce
+  have "\<And>i. i \<in> I \<Longrightarrow> openin X (T i \<inter> f i -` U)"
+    using cont unfolding continuous_map_openin_preimage_eq
+    by (metis Y T inf.commute inf_absorb1 ope topspace_subtopology openin_trans_full)
+  then show "openin X (topspace X \<inter> g -` U)"
+    by (auto simp: *)
 qed
 
 lemma pasting_lemma_exists:
-  fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
-  assumes S: "S \<subseteq> (\<Union>i \<in> I. T i)"
-      and clo: "\<And>i. i \<in> I \<Longrightarrow> openin (subtopology euclidean S) (T i)"
-      and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
-      and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
-    obtains g where "continuous_on S g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> S \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
+  assumes X: "topspace X \<subseteq> (\<Union>i \<in> I. T i)"
+      and ope: "\<And>i. i \<in> I \<Longrightarrow> openin X (T i)"
+      and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_map (subtopology X (T i)) Y (f i)"
+      and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
+    obtains g where "continuous_map X Y g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> topspace X \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
 proof
-  show "continuous_on S (\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x)"
-    apply (rule pasting_lemma [OF clo cont])
+  let ?h = "\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x"
+  show "continuous_map X Y ?h"
+    apply (rule pasting_lemma [OF ope cont])
      apply (blast intro: f)+
-    apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)
+    by (metis (no_types, lifting) UN_E X subsetD someI_ex)
+  show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x" if "i \<in> I" "x \<in> topspace X \<inter> T i" for i x
+    by (metis (no_types, lifting) IntD2 IntI f someI_ex that)
+qed
+
+lemma pasting_lemma_locally_finite:
+  assumes fin: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>V. openin X V \<and> x \<in> V \<and> finite {i \<in> I. T i \<inter> V \<noteq> {}}"
+    and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i)"
+    and cont:  "\<And>i. i \<in> I \<Longrightarrow> continuous_map(subtopology X (T i)) Y (f i)"
+    and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
+    and g: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
+  shows "continuous_map X Y g"
+  unfolding continuous_map_closedin_preimage_eq
+proof (intro conjI allI impI)
+  show "g ` topspace X \<subseteq> topspace Y"
+    using g cont continuous_map_image_subset_topspace topspace_subtopology by fastforce
+next
+  fix U
+  assume Y: "closedin Y U"
+  have T: "T i \<subseteq> topspace X" if "i \<in> I" for i
+    using clo by (simp add: closedin_subset that)
+  have *: "topspace X \<inter> g -` U = (\<Union>i \<in> I. T i \<inter> f i -` U)"
+    using f g T by fastforce
+  have cTf: "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i \<inter> f i -` U)"
+    using cont unfolding continuous_map_closedin_preimage_eq topspace_subtopology
+    by (simp add: Int_absorb1 T Y clo closedin_closed_subtopology)
+  have sub: "{Z \<in> (\<lambda>i. T i \<inter> f i -` U) ` I. Z \<inter> V \<noteq> {}}
+           \<subseteq> (\<lambda>i. T i \<inter> f i -` U) ` {i \<in> I. T i \<inter> V \<noteq> {}}" for V
+    by auto
+  have 1: "(\<Union>i\<in>I. T i \<inter> f i -` U) \<subseteq> topspace X"
+    using T by blast
+  then have lf: "locally_finite_in X ((\<lambda>i. T i \<inter> f i -` U) ` I)"
+    unfolding locally_finite_in_def
+    using finite_subset [OF sub] fin by force
+  show "closedin X (topspace X \<inter> g -` U)"
+    apply (subst *)
+    apply (rule closedin_locally_finite_Union)
+     apply (auto intro: cTf lf)
     done
+qed
+
+subsubsection\<open>Likewise on closed sets, with a finiteness assumption\<close>
+
+lemma pasting_lemma_closed:
+  assumes fin: "finite I"
+    and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i)"
+    and cont:  "\<And>i. i \<in> I \<Longrightarrow> continuous_map(subtopology X (T i)) Y (f i)"
+    and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
+    and g: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
+  shows "continuous_map X Y g"
+  using pasting_lemma_locally_finite [OF _ clo cont f g] fin by auto
+
+lemma pasting_lemma_exists_locally_finite:
+  assumes fin: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>V. openin X V \<and> x \<in> V \<and> finite {i \<in> I. T i \<inter> V \<noteq> {}}"
+    and X: "topspace X \<subseteq> \<Union>(T ` I)"
+    and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i)"
+    and cont:  "\<And>i. i \<in> I \<Longrightarrow> continuous_map(subtopology X (T i)) Y (f i)"
+    and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
+    and g: "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
+  obtains g where "continuous_map X Y g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> topspace X \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
+proof
+  show "continuous_map X Y (\<lambda>x. f(@i. i \<in> I \<and> x \<in> T i) x)"
+    apply (rule pasting_lemma_locally_finite [OF fin])
+        apply (blast intro: assms)+
+    by (metis (no_types, lifting) UN_E X set_rev_mp someI_ex)
 next
   fix x i
-  assume "i \<in> I" "x \<in> S \<inter> T i"
+  assume "i \<in> I" and "x \<in> topspace X \<inter> T i"
+  show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x"
+    apply (rule someI2_ex)
+    using \<open>i \<in> I\<close> \<open>x \<in> topspace X \<inter> T i\<close> apply blast
+    by (meson Int_iff \<open>i \<in> I\<close> \<open>x \<in> topspace X \<inter> T i\<close> f)
+qed
+
+lemma pasting_lemma_exists_closed:
+  assumes fin: "finite I"
+    and X: "topspace X \<subseteq> \<Union>(T ` I)"
+    and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i)"
+    and cont:  "\<And>i. i \<in> I \<Longrightarrow> continuous_map(subtopology X (T i)) Y (f i)"
+    and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
+  obtains g where "continuous_map X Y g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> topspace X \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
+proof
+  show "continuous_map X Y (\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x)"
+    apply (rule pasting_lemma_closed [OF \<open>finite I\<close> clo cont])
+     apply (blast intro: f)+
+    by (metis (mono_tags, lifting) UN_iff X someI_ex subset_iff)
+next
+  fix x i
+  assume "i \<in> I" "x \<in> topspace X \<inter> T i"
   then show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x"
     by (metis (no_types, lifting) IntD2 IntI f someI_ex)
 qed
 
-text\<open>Likewise on closed sets, with a finiteness assumption\<close>
+lemma continuous_map_cases:
+  assumes f: "continuous_map (subtopology X (X closure_of {x. P x})) Y f"
+      and g: "continuous_map (subtopology X (X closure_of {x. \<not> P x})) Y g"
+      and fg: "\<And>x. x \<in> X frontier_of {x. P x} \<Longrightarrow> f x = g x"
+  shows "continuous_map X Y (\<lambda>x. if P x then f x else g x)"
+proof (rule pasting_lemma_closed)
+  let ?f = "\<lambda>b. if b then f else g"
+  let ?g = "\<lambda>x. if P x then f x else g x"
+  let ?T = "\<lambda>b. if b then X closure_of {x. P x} else X closure_of {x. ~P x}"
+  show "finite {True,False}" by auto
+  have eq: "topspace X - Collect P = topspace X \<inter> {x. \<not> P x}"
+    by blast
+  show "?f i x = ?f j x"
+    if "i \<in> {True,False}" "j \<in> {True,False}" and x: "x \<in> topspace X \<inter> ?T i \<inter> ?T j" for i j x
+  proof -
+    have "f x = g x"
+      if "i" "\<not> j"
+      apply (rule fg)
+      unfolding frontier_of_closures eq
+      using x that closure_of_restrict by fastforce
+    moreover
+    have "g x = f x"
+      if "x \<in> X closure_of {x. \<not> P x}" "x \<in> X closure_of Collect P" "\<not> i" "j" for x
+        apply (rule fg [symmetric])
+        unfolding frontier_of_closures eq
+        using x that closure_of_restrict by fastforce
+    ultimately show ?thesis
+      using that by (auto simp flip: closure_of_restrict)
+  qed
+  show "\<exists>j. j \<in> {True,False} \<and> x \<in> ?T j \<and> (if P x then f x else g x) = ?f j x"
+    if "x \<in> topspace X" for x
+    apply simp
+    apply safe
+    apply (metis Int_iff closure_of inf_sup_absorb mem_Collect_eq that)
+    by (metis DiffI eq closure_of_subset_Int contra_subsetD mem_Collect_eq that)
+qed (auto simp: f g)
 
-lemma pasting_lemma_closed:
-  fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
-  assumes "finite I"
-      and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin (subtopology euclidean S) (T i)"
-      and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
-      and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
-      and g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
-    shows "continuous_on S g"
-proof (clarsimp simp: continuous_closedin_preimage_eq)
-  fix U :: "'b set"
-  assume "closed U"
-  have S: "\<And>i. i \<in> I \<Longrightarrow> (T i) \<subseteq> S"
-    using clo closedin_imp_subset by blast
-  have *: "(S \<inter> g -` U) = (\<Union>i \<in> I. T i \<inter> f i -` U)"
-    using S f g by fastforce
-  show "closedin (subtopology euclidean S) (S \<inter> g -` U)"
-    apply (subst *)
-    apply (rule closedin_Union)
-    using \<open>finite I\<close> apply simp
-    apply (blast intro: \<open>closed U\<close> continuous_closedin_preimage cont clo closedin_trans)
-    done
+lemma continuous_map_cases_alt:
+  assumes f: "continuous_map (subtopology X (X closure_of {x \<in> topspace X. P x})) Y f"
+      and g: "continuous_map (subtopology X (X closure_of {x \<in> topspace X. ~P x})) Y g"
+      and fg: "\<And>x. x \<in> X frontier_of {x \<in> topspace X. P x} \<Longrightarrow> f x = g x"
+    shows "continuous_map X Y (\<lambda>x. if P x then f x else g x)"
+  apply (rule continuous_map_cases)
+  using assms
+    apply (simp_all add: Collect_conj_eq closure_of_restrict [symmetric] frontier_of_restrict [symmetric])
+  done
+
+lemma continuous_map_cases_function:
+  assumes contp: "continuous_map X Z p"
+    and contf: "continuous_map (subtopology X {x \<in> topspace X. p x \<in> Z closure_of U}) Y f"
+    and contg: "continuous_map (subtopology X {x \<in> topspace X. p x \<in> Z closure_of (topspace Z - U)}) Y g"
+    and fg: "\<And>x. \<lbrakk>x \<in> topspace X; p x \<in> Z frontier_of U\<rbrakk> \<Longrightarrow> f x = g x"
+  shows "continuous_map X Y (\<lambda>x. if p x \<in> U then f x else g x)"
+proof (rule continuous_map_cases_alt)
+  show "continuous_map (subtopology X (X closure_of {x \<in> topspace X. p x \<in> U})) Y f"
+  proof (rule continuous_map_from_subtopology_mono)
+    let ?T = "{x \<in> topspace X. p x \<in> Z closure_of U}"
+    show "continuous_map (subtopology X ?T) Y f"
+      by (simp add: contf)
+    show "X closure_of {x \<in> topspace X. p x \<in> U} \<subseteq> ?T"
+      by (rule continuous_map_closure_preimage_subset [OF contp])
+  qed
+  show "continuous_map (subtopology X (X closure_of {x \<in> topspace X. p x \<notin> U})) Y g"
+  proof (rule continuous_map_from_subtopology_mono)
+    let ?T = "{x \<in> topspace X. p x \<in> Z closure_of (topspace Z - U)}"
+    show "continuous_map (subtopology X ?T) Y g"
+      by (simp add: contg)
+    have "X closure_of {x \<in> topspace X. p x \<notin> U} \<subseteq> X closure_of {x \<in> topspace X. p x \<in> topspace Z - U}"
+      apply (rule closure_of_mono)
+      using continuous_map_closedin contp by fastforce
+    then show "X closure_of {x \<in> topspace X. p x \<notin> U} \<subseteq> ?T"
+      by (rule order_trans [OF _ continuous_map_closure_preimage_subset [OF contp]])
+  qed
+next
+  show "f x = g x" if "x \<in> X frontier_of {x \<in> topspace X. p x \<in> U}" for x
+    using that continuous_map_frontier_frontier_preimage_subset [OF contp, of U] fg by blast
 qed
 
-lemma pasting_lemma_exists_closed:
-  fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
-  assumes "finite I"
-      and S: "S \<subseteq> (\<Union>i \<in> I. T i)"
-      and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin (subtopology euclidean S) (T i)"
-      and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
-      and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
-    obtains g where "continuous_on S g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> S \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
-proof
-  show "continuous_on S (\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x)"
-    apply (rule pasting_lemma_closed [OF \<open>finite I\<close> clo cont])
-     apply (blast intro: f)+
-    apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)
-    done
-next
-  fix x i
-  assume "i \<in> I" "x \<in> S \<inter> T i"
-  then show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x"
-    by (metis (no_types, lifting) IntD2 IntI f someI_ex)
-qed
-
-
 subsection \<open>Retractions\<close>
 
 definition%important retraction :: "('a::topological_space) set \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool"
@@ -948,7 +1076,7 @@
 lemma closedin_retract:
   fixes S :: "'a :: t2_space set"
   assumes "S retract_of T"
-    shows "closedin (subtopology euclidean T) S"
+    shows "closedin (top_of_set T) S"
 proof -
   obtain r where r: "S \<subseteq> T" "continuous_on T r" "r ` T \<subseteq> S" "\<And>x. x \<in> S \<Longrightarrow> r x = x"
     using assms by (auto simp: retract_of_def retraction_def)
@@ -963,7 +1091,7 @@
   finally show ?thesis .
 qed
 
-lemma closedin_self [simp]: "closedin (subtopology euclidean S) S"
+lemma closedin_self [simp]: "closedin (top_of_set S) S"
   by simp
 
 lemma retract_of_closed:
@@ -980,7 +1108,7 @@
   by (metis Topological_Spaces.connected_continuous_image retract_of_def retraction)
 
 lemma retraction_imp_quotient_map:
-  "openin (subtopology euclidean S) (S \<inter> r -` U) \<longleftrightarrow> openin (subtopology euclidean T) U"
+  "openin (top_of_set S) (S \<inter> r -` U) \<longleftrightarrow> openin (top_of_set T) U"
   if retraction: "retraction S T r" and "U \<subseteq> T"
   using retraction apply (rule retractionE)
   apply (rule continuous_right_inverse_imp_quotient_map [where g=r])
@@ -995,4 +1123,472 @@
 apply (rule conjI continuous_intros | erule continuous_on_subset | force)+
 done
 
+subsection\<open>Retractions on a topological space\<close>
+
+definition retract_of_space :: "'a set \<Rightarrow> 'a topology \<Rightarrow> bool" (infix "retract'_of'_space" 50)
+  where "S retract_of_space X
+         \<equiv> S \<subseteq> topspace X \<and> (\<exists>r. continuous_map X (subtopology X S) r \<and> (\<forall>x \<in> S. r x = x))"
+
+lemma retract_of_space_retraction_maps:
+   "S retract_of_space X \<longleftrightarrow> S \<subseteq> topspace X \<and> (\<exists>r. retraction_maps X (subtopology X S) r id)"
+  by (auto simp: retract_of_space_def retraction_maps_def)
+
+lemma retract_of_space_section_map:
+   "S retract_of_space X \<longleftrightarrow> S \<subseteq> topspace X \<and> section_map (subtopology X S) X id"
+  unfolding retract_of_space_def retraction_maps_def section_map_def
+  by (auto simp: continuous_map_from_subtopology)
+
+lemma retract_of_space_imp_subset:
+   "S retract_of_space X \<Longrightarrow> S \<subseteq> topspace X"
+  by (simp add: retract_of_space_def)
+
+lemma retract_of_space_topspace:
+   "topspace X retract_of_space X"
+  using retract_of_space_def by force
+
+lemma retract_of_space_empty [simp]:
+   "{} retract_of_space X \<longleftrightarrow> topspace X = {}"
+  by (auto simp: continuous_map_def retract_of_space_def)
+
+lemma retract_of_space_singleton [simp]:
+  "{a} retract_of_space X \<longleftrightarrow> a \<in> topspace X"
+proof -
+  have "continuous_map X (subtopology X {a}) (\<lambda>x. a) \<and> (\<lambda>x. a) a = a" if "a \<in> topspace X"
+    using that by simp
+  then show ?thesis
+    by (force simp: retract_of_space_def)
+qed
+
+lemma retract_of_space_clopen:
+  assumes "openin X S" "closedin X S" "S = {} \<Longrightarrow> topspace X = {}"
+  shows "S retract_of_space X"
+proof (cases "S = {}")
+  case False
+  then obtain a where "a \<in> S"
+    by blast
+  show ?thesis
+    unfolding retract_of_space_def
+  proof (intro exI conjI)
+    show "S \<subseteq> topspace X"
+      by (simp add: assms closedin_subset)
+    have "continuous_map X X (\<lambda>x. if x \<in> S then x else a)"
+    proof (rule continuous_map_cases)
+      show "continuous_map (subtopology X (X closure_of {x. x \<in> S})) X (\<lambda>x. x)"
+        by (simp add: continuous_map_from_subtopology)
+      show "continuous_map (subtopology X (X closure_of {x. x \<notin> S})) X (\<lambda>x. a)"
+        using \<open>S \<subseteq> topspace X\<close> \<open>a \<in> S\<close> by force
+      show "x = a" if "x \<in> X frontier_of {x. x \<in> S}" for x
+        using assms that clopenin_eq_frontier_of by fastforce
+    qed
+    then show "continuous_map X (subtopology X S) (\<lambda>x. if x \<in> S then x else a)"
+      using \<open>S \<subseteq> topspace X\<close> \<open>a \<in> S\<close>  by (auto simp: continuous_map_in_subtopology)
+  qed auto
+qed (use assms in auto)
+
+lemma retract_of_space_disjoint_union:
+  assumes "openin X S" "openin X T" and ST: "disjnt S T" "S \<union> T = topspace X" and "S = {} \<Longrightarrow> topspace X = {}"
+  shows "S retract_of_space X"
+proof (rule retract_of_space_clopen)
+  have "S \<inter> T = {}"
+    by (meson ST disjnt_def)
+  then have "S = topspace X - T"
+    using ST by auto
+  then show "closedin X S"
+    using \<open>openin X T\<close> by blast
+qed (auto simp: assms)
+
+lemma retraction_maps_section_image1:
+  assumes "retraction_maps X Y r s"
+  shows "s ` (topspace Y) retract_of_space X"
+  unfolding retract_of_space_section_map
+proof
+  show "s ` topspace Y \<subseteq> topspace X"
+    using assms continuous_map_image_subset_topspace retraction_maps_def by blast
+  show "section_map (subtopology X (s ` topspace Y)) X id"
+    unfolding section_map_def
+    using assms retraction_maps_to_retract_maps by blast
+qed
+
+lemma retraction_maps_section_image2:
+   "retraction_maps X Y r s
+        \<Longrightarrow> subtopology X (s ` (topspace Y)) homeomorphic_space Y"
+  using embedding_map_imp_homeomorphic_space homeomorphic_space_sym section_imp_embedding_map
+        section_map_def by blast
+
+subsection\<open>Paths and path-connectedness\<close>
+
+definition pathin :: "'a topology \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> bool" where
+   "pathin X g \<equiv> continuous_map (subtopology euclideanreal {0..1}) X g"
+
+lemma pathin_compose:
+     "\<lbrakk>pathin X g; continuous_map X Y f\<rbrakk> \<Longrightarrow> pathin Y (f \<circ> g)"
+   by (simp add: continuous_map_compose pathin_def)
+
+lemma pathin_subtopology:
+     "pathin (subtopology X S) g \<longleftrightarrow> pathin X g \<and> (\<forall>x \<in> {0..1}. g x \<in> S)"
+  by (auto simp: pathin_def continuous_map_in_subtopology)
+
+lemma pathin_const:
+   "pathin X (\<lambda>x. a) \<longleftrightarrow> a \<in> topspace X"
+  by (simp add: pathin_def)
+
+definition path_connected_space :: "'a topology \<Rightarrow> bool"
+  where "path_connected_space X \<equiv> \<forall>x \<in> topspace X. \<forall> y \<in> topspace X. \<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y"
+
+definition path_connectedin :: "'a topology \<Rightarrow> 'a set \<Rightarrow> bool"
+  where "path_connectedin X S \<equiv> S \<subseteq> topspace X \<and> path_connected_space(subtopology X S)"
+
+lemma path_connectedin_absolute [simp]:
+     "path_connectedin (subtopology X S) S \<longleftrightarrow> path_connectedin X S"
+  by (simp add: path_connectedin_def subtopology_subtopology topspace_subtopology)
+
+lemma path_connectedin_subset_topspace:
+     "path_connectedin X S \<Longrightarrow> S \<subseteq> topspace X"
+  by (simp add: path_connectedin_def)
+
+lemma path_connectedin_subtopology:
+     "path_connectedin (subtopology X S) T \<longleftrightarrow> path_connectedin X T \<and> T \<subseteq> S"
+  by (auto simp: path_connectedin_def subtopology_subtopology topspace_subtopology inf.absorb2)
+
+lemma path_connectedin:
+     "path_connectedin X S \<longleftrightarrow>
+        S \<subseteq> topspace X \<and>
+        (\<forall>x \<in> S. \<forall>y \<in> S. \<exists>g. pathin X g \<and> g ` {0..1} \<subseteq> S \<and> g 0 = x \<and> g 1 = y)"
+  unfolding path_connectedin_def path_connected_space_def pathin_def continuous_map_in_subtopology
+  by (intro conj_cong refl ball_cong) (simp_all add: inf.absorb_iff2 topspace_subtopology)
+
+lemma path_connectedin_topspace:
+     "path_connectedin X (topspace X) \<longleftrightarrow> path_connected_space X"
+  by (simp add: path_connectedin_def)
+
+lemma path_connected_imp_connected_space:
+  assumes "path_connected_space X"
+  shows "connected_space X"
+proof -
+  have *: "\<exists>S. connectedin X S \<and> g 0 \<in> S \<and> g 1 \<in> S" if "pathin X g" for g
+  proof (intro exI conjI)
+    have "continuous_map (subtopology euclideanreal {0..1}) X g"
+      using connectedin_absolute that by (simp add: pathin_def)
+    then show "connectedin X (g ` {0..1})"
+      by (rule connectedin_continuous_map_image) auto
+  qed auto
+  show ?thesis
+    using assms
+    by (auto intro: * simp add: path_connected_space_def connected_space_subconnected Ball_def)
+qed
+
+lemma path_connectedin_imp_connectedin:
+     "path_connectedin X S \<Longrightarrow> connectedin X S"
+  by (simp add: connectedin_def path_connected_imp_connected_space path_connectedin_def)
+
+lemma path_connected_space_topspace_empty:
+     "topspace X = {} \<Longrightarrow> path_connected_space X"
+  by (simp add: path_connected_space_def)
+
+lemma path_connectedin_empty [simp]: "path_connectedin X {}"
+  by (simp add: path_connectedin)
+
+lemma path_connectedin_singleton [simp]: "path_connectedin X {a} \<longleftrightarrow> a \<in> topspace X"
+proof
+  show "path_connectedin X {a} \<Longrightarrow> a \<in> topspace X"
+    by (simp add: path_connectedin)
+  show "a \<in> topspace X \<Longrightarrow> path_connectedin X {a}"
+    unfolding path_connectedin
+    using pathin_const by fastforce
+qed
+
+lemma path_connectedin_continuous_map_image:
+  assumes f: "continuous_map X Y f" and S: "path_connectedin X S"
+  shows "path_connectedin Y (f ` S)"
+proof -
+  have fX: "f ` (topspace X) \<subseteq> topspace Y"
+    by (metis f continuous_map_image_subset_topspace)
+  show ?thesis
+    unfolding path_connectedin
+  proof (intro conjI ballI; clarify?)
+    fix x
+    assume "x \<in> S"
+    show "f x \<in> topspace Y"
+      by (meson S fX \<open>x \<in> S\<close> image_subset_iff path_connectedin_subset_topspace set_mp)
+  next
+    fix x y
+    assume "x \<in> S" and "y \<in> S"
+    then obtain g where g: "pathin X g" "g ` {0..1} \<subseteq> S" "g 0 = x" "g 1 = y"
+      using S  by (force simp: path_connectedin)
+    show "\<exists>g. pathin Y g \<and> g ` {0..1} \<subseteq> f ` S \<and> g 0 = f x \<and> g 1 = f y"
+    proof (intro exI conjI)
+      show "pathin Y (f \<circ> g)"
+        using \<open>pathin X g\<close> f pathin_compose by auto
+    qed (use g in auto)
+  qed
+qed
+
+lemma homeomorphic_path_connected_space_imp:
+     "\<lbrakk>path_connected_space X; X homeomorphic_space Y\<rbrakk> \<Longrightarrow> path_connected_space Y"
+  unfolding homeomorphic_space_def homeomorphic_maps_def
+  by (metis (no_types, hide_lams) continuous_map_closedin continuous_map_image_subset_topspace imageI order_class.order.antisym path_connectedin_continuous_map_image path_connectedin_topspace subsetI)
+
+lemma homeomorphic_path_connected_space:
+   "X homeomorphic_space Y \<Longrightarrow> path_connected_space X \<longleftrightarrow> path_connected_space Y"
+  by (meson homeomorphic_path_connected_space_imp homeomorphic_space_sym)
+
+lemma homeomorphic_map_path_connectedness:
+  assumes "homeomorphic_map X Y f" "U \<subseteq> topspace X"
+  shows "path_connectedin Y (f ` U) \<longleftrightarrow> path_connectedin X U"
+  unfolding path_connectedin_def
+proof (intro conj_cong homeomorphic_path_connected_space)
+  show "(f ` U \<subseteq> topspace Y) = (U \<subseteq> topspace X)"
+    using assms homeomorphic_imp_surjective_map by blast
+next
+  assume "U \<subseteq> topspace X"
+  show "subtopology Y (f ` U) homeomorphic_space subtopology X U"
+    using assms unfolding homeomorphic_eq_everything_map
+    by (metis (no_types, hide_lams) assms homeomorphic_map_subtopologies homeomorphic_space homeomorphic_space_sym image_mono inf.absorb_iff2)
+qed
+
+lemma homeomorphic_map_path_connectedness_eq:
+   "homeomorphic_map X Y f \<Longrightarrow> path_connectedin X U \<longleftrightarrow> U \<subseteq> topspace X \<and> path_connectedin Y (f ` U)"
+  by (meson homeomorphic_map_path_connectedness path_connectedin_def)
+
+subsection\<open>Connected components\<close>
+
+definition connected_component_of :: "'a topology \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
+  where "connected_component_of X x y \<equiv>
+        \<exists>T. connectedin X T \<and> x \<in> T \<and> y \<in> T"
+
+abbreviation connected_component_of_set
+  where "connected_component_of_set X x \<equiv> Collect (connected_component_of X x)"
+
+definition connected_components_of :: "'a topology \<Rightarrow> ('a set) set"
+  where "connected_components_of X \<equiv> connected_component_of_set X ` topspace X"
+
+lemma connected_component_in_topspace:
+   "connected_component_of X x y \<Longrightarrow> x \<in> topspace X \<and> y \<in> topspace X"
+  by (meson connected_component_of_def connectedin_subset_topspace in_mono)
+
+lemma connected_component_of_refl:
+   "connected_component_of X x x \<longleftrightarrow> x \<in> topspace X"
+  by (meson connected_component_in_topspace connected_component_of_def connectedin_sing insertI1)
+
+lemma connected_component_of_sym:
+   "connected_component_of X x y \<longleftrightarrow> connected_component_of X y x"
+  by (meson connected_component_of_def)
+
+lemma connected_component_of_trans:
+   "\<lbrakk>connected_component_of X x y; connected_component_of X y z\<rbrakk>
+        \<Longrightarrow> connected_component_of X x z"
+  unfolding connected_component_of_def
+  using connectedin_Un by fastforce
+
+lemma connected_component_of_mono:
+   "\<lbrakk>connected_component_of (subtopology X S) x y; S \<subseteq> T\<rbrakk>
+        \<Longrightarrow> connected_component_of (subtopology X T) x y"
+  by (metis connected_component_of_def connectedin_subtopology inf.absorb_iff2 subtopology_subtopology)
+
+lemma connected_component_of_set:
+   "connected_component_of_set X x = {y. \<exists>T. connectedin X T \<and> x \<in> T \<and> y \<in> T}"
+  by (meson connected_component_of_def)
+
+lemma connected_component_of_subset_topspace:
+   "connected_component_of_set X x \<subseteq> topspace X"
+  using connected_component_in_topspace by force
+
+lemma connected_component_of_eq_empty:
+   "connected_component_of_set X x = {} \<longleftrightarrow> (x \<notin> topspace X)"
+  using connected_component_in_topspace connected_component_of_refl by fastforce
+
+lemma connected_space_iff_connected_component:
+   "connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. connected_component_of X x y)"
+  by (simp add: connected_component_of_def connected_space_subconnected)
+
+lemma connected_space_imp_connected_component_of:
+   "\<lbrakk>connected_space X; a \<in> topspace X; b \<in> topspace X\<rbrakk>
+    \<Longrightarrow> connected_component_of X a b"
+  by (simp add: connected_space_iff_connected_component)
+
+lemma connected_space_connected_component_set:
+   "connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. connected_component_of_set X x = topspace X)"
+  using connected_component_of_subset_topspace connected_space_iff_connected_component by fastforce
+
+lemma connected_component_of_maximal:
+   "\<lbrakk>connectedin X S; x \<in> S\<rbrakk> \<Longrightarrow> S \<subseteq> connected_component_of_set X x"
+  by (meson Ball_Collect connected_component_of_def)
+
+lemma connected_component_of_equiv:
+   "connected_component_of X x y \<longleftrightarrow>
+    x \<in> topspace X \<and> y \<in> topspace X \<and> connected_component_of X x = connected_component_of X y"
+  apply (simp add: connected_component_in_topspace fun_eq_iff)
+  by (meson connected_component_of_refl connected_component_of_sym connected_component_of_trans)
+
+lemma connected_component_of_disjoint:
+   "disjnt (connected_component_of_set X x) (connected_component_of_set X y)
+    \<longleftrightarrow> ~(connected_component_of X x y)"
+  using connected_component_of_equiv unfolding disjnt_iff by force
+
+lemma connected_component_of_eq:
+   "connected_component_of X x = connected_component_of X y \<longleftrightarrow>
+        (x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or>
+        x \<in> topspace X \<and> y \<in> topspace X \<and>
+        connected_component_of X x y"
+  by (metis Collect_empty_eq_bot connected_component_of_eq_empty connected_component_of_equiv)
+
+lemma connectedin_connected_component_of:
+   "connectedin X (connected_component_of_set X x)"
+proof -
+  have "connected_component_of_set X x = \<Union> {T. connectedin X T \<and> x \<in> T}"
+    by (auto simp: connected_component_of_def)
+  then show ?thesis
+    apply (rule ssubst)
+    by (blast intro: connectedin_Union)
+qed
+
+
+lemma Union_connected_components_of:
+   "\<Union>(connected_components_of X) = topspace X"
+  unfolding connected_components_of_def
+  apply (rule equalityI)
+  apply (simp add: SUP_least connected_component_of_subset_topspace)
+  using connected_component_of_refl by fastforce
+
+lemma connected_components_of_maximal:
+   "\<lbrakk>C \<in> connected_components_of X; connectedin X S; ~disjnt C S\<rbrakk> \<Longrightarrow> S \<subseteq> C"
+  unfolding connected_components_of_def disjnt_def
+  apply clarify
+  by (metis Int_emptyI connected_component_of_def connected_component_of_trans mem_Collect_eq)
+
+lemma pairwise_disjoint_connected_components_of:
+   "pairwise disjnt (connected_components_of X)"
+  unfolding connected_components_of_def pairwise_def
+  apply clarify
+  by (metis connected_component_of_disjoint connected_component_of_equiv)
+
+lemma complement_connected_components_of_Union:
+   "C \<in> connected_components_of X
+      \<Longrightarrow> topspace X - C = \<Union> (connected_components_of X - {C})"
+  apply (rule equalityI)
+  using Union_connected_components_of apply fastforce
+  by (metis Diff_cancel Diff_subset Union_connected_components_of cSup_singleton diff_Union_pairwise_disjoint equalityE insert_subsetI pairwise_disjoint_connected_components_of)
+
+lemma nonempty_connected_components_of:
+   "C \<in> connected_components_of X \<Longrightarrow> C \<noteq> {}"
+  unfolding connected_components_of_def
+  by (metis (no_types, lifting) connected_component_of_eq_empty imageE)
+
+lemma connected_components_of_subset:
+   "C \<in> connected_components_of X \<Longrightarrow> C \<subseteq> topspace X"
+  using Union_connected_components_of by fastforce
+
+lemma connectedin_connected_components_of:
+  assumes "C \<in> connected_components_of X"
+  shows "connectedin X C"
+proof -
+  have "C \<in> connected_component_of_set X ` topspace X"
+    using assms connected_components_of_def by blast
+then show ?thesis
+  using connectedin_connected_component_of by fastforce
+qed
+
+lemma connected_component_in_connected_components_of:
+   "connected_component_of_set X a \<in> connected_components_of X \<longleftrightarrow> a \<in> topspace X"
+  apply (rule iffI)
+  using connected_component_of_eq_empty nonempty_connected_components_of apply fastforce
+  by (simp add: connected_components_of_def)
+
+lemma connected_space_iff_components_eq:
+   "connected_space X \<longleftrightarrow> (\<forall>C \<in> connected_components_of X. \<forall>C' \<in> connected_components_of X. C = C')"
+  apply (rule iffI)
+  apply (force simp: connected_components_of_def connected_space_connected_component_set image_iff)
+  by (metis connected_component_in_connected_components_of connected_component_of_refl connected_space_iff_connected_component mem_Collect_eq)
+
+lemma connected_components_of_eq_empty:
+   "connected_components_of X = {} \<longleftrightarrow> topspace X = {}"
+  by (simp add: connected_components_of_def)
+
+lemma connected_components_of_empty_space:
+   "topspace X = {} \<Longrightarrow> connected_components_of X = {}"
+  by (simp add: connected_components_of_eq_empty)
+
+lemma connected_components_of_subset_sing:
+   "connected_components_of X \<subseteq> {S} \<longleftrightarrow> connected_space X \<and> (topspace X = {} \<or> topspace X = S)"
+proof (cases "topspace X = {}")
+  case True
+  then show ?thesis
+    by (simp add: connected_components_of_empty_space connected_space_topspace_empty)
+next
+  case False
+  then show ?thesis
+    by (metis (no_types, hide_lams) Union_connected_components_of ccpo_Sup_singleton
+        connected_components_of_eq_empty connected_space_iff_components_eq insertI1 singletonD
+        subsetI subset_singleton_iff)
+qed
+
+lemma connected_space_iff_components_subset_singleton:
+   "connected_space X \<longleftrightarrow> (\<exists>a. connected_components_of X \<subseteq> {a})"
+  by (simp add: connected_components_of_subset_sing)
+
+lemma connected_components_of_eq_singleton:
+   "connected_components_of X = {S}
+\<longleftrightarrow> connected_space X \<and> topspace X \<noteq> {} \<and> S = topspace X"
+  by (metis ccpo_Sup_singleton connected_components_of_subset_sing insert_not_empty subset_singleton_iff)
+
+lemma connected_components_of_connected_space:
+   "connected_space X \<Longrightarrow> connected_components_of X = (if topspace X = {} then {} else {topspace X})"
+  by (simp add: connected_components_of_eq_empty connected_components_of_eq_singleton)
+
+lemma exists_connected_component_of_superset:
+  assumes "connectedin X S" and ne: "topspace X \<noteq> {}"
+  shows "\<exists>C. C \<in> connected_components_of X \<and> S \<subseteq> C"
+proof (cases "S = {}")
+  case True
+  then show ?thesis
+    using ne connected_components_of_def by blast
+next
+  case False
+  then show ?thesis
+    by (meson all_not_in_conv assms(1) connected_component_in_connected_components_of connected_component_of_maximal connectedin_subset_topspace in_mono)
+qed
+
+lemma closedin_connected_components_of:
+  assumes "C \<in> connected_components_of X"
+  shows   "closedin X C"
+proof -
+  obtain x where "x \<in> topspace X" and x: "C = connected_component_of_set X x"
+    using assms by (auto simp: connected_components_of_def)
+  have "connected_component_of_set X x \<subseteq> topspace X"
+    by (simp add: connected_component_of_subset_topspace)
+  moreover have "X closure_of connected_component_of_set X x \<subseteq> connected_component_of_set X x"
+  proof (rule connected_component_of_maximal)
+    show "connectedin X (X closure_of connected_component_of_set X x)"
+      by (simp add: connectedin_closure_of connectedin_connected_component_of)
+    show "x \<in> X closure_of connected_component_of_set X x"
+      by (simp add: \<open>x \<in> topspace X\<close> closure_of connected_component_of_refl)
+  qed
+  ultimately
+  show ?thesis
+    using closure_of_subset_eq x by auto
+qed
+
+lemma closedin_connected_component_of:
+   "closedin X (connected_component_of_set X x)"
+  by (metis closedin_connected_components_of closedin_empty connected_component_in_connected_components_of connected_component_of_eq_empty)
+
+lemma connected_component_of_eq_overlap:
+   "connected_component_of_set X x = connected_component_of_set X y \<longleftrightarrow>
+      (x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or>
+      ~(connected_component_of_set X x \<inter> connected_component_of_set X y = {})"
+  using connected_component_of_equiv by fastforce
+
+lemma connected_component_of_nonoverlap:
+   "connected_component_of_set X x \<inter> connected_component_of_set X y = {} \<longleftrightarrow>
+     (x \<notin> topspace X) \<or> (y \<notin> topspace X) \<or>
+     ~(connected_component_of_set X x = connected_component_of_set X y)"
+  by (metis connected_component_of_eq_empty connected_component_of_eq_overlap inf.idem)
+
+lemma connected_component_of_overlap:
+   "~(connected_component_of_set X x \<inter> connected_component_of_set X y = {}) \<longleftrightarrow>
+    x \<in> topspace X \<and> y \<in> topspace X \<and>
+    connected_component_of_set X x = connected_component_of_set X y"
+  by (meson connected_component_of_nonoverlap)
+
+
 end
\ No newline at end of file
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Analysis.thy
--- a/src/HOL/Analysis/Analysis.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Analysis.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -27,7 +27,7 @@
   Homeomorphism
   Bounded_Continuous_Function
   Abstract_Topology
-  Function_Topology
+  Product_Topology
   T1_Spaces
   Infinite_Products
   Infinite_Set_Sum
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Arcwise_Connected.thy
--- a/src/HOL/Analysis/Arcwise_Connected.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Arcwise_Connected.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -2005,7 +2005,7 @@
 
 lemma dense_accessible_frontier_points:
   fixes S :: "'a::{complete_space,real_normed_vector} set"
-  assumes "open S" and opeSV: "openin (subtopology euclidean (frontier S)) V" and "V \<noteq> {}"
+  assumes "open S" and opeSV: "openin (top_of_set (frontier S)) V" and "V \<noteq> {}"
   obtains g where "arc g" "g ` {0..<1} \<subseteq> S" "pathstart g \<in> S" "pathfinish g \<in> V"
 proof -
   obtain z where "z \<in> V"
@@ -2101,7 +2101,7 @@
 lemma dense_accessible_frontier_points_connected:
   fixes S :: "'a::{complete_space,real_normed_vector} set"
   assumes "open S" "connected S" "x \<in> S" "V \<noteq> {}"
-      and ope: "openin (subtopology euclidean (frontier S)) V"
+      and ope: "openin (top_of_set (frontier S)) V"
   obtains g where "arc g" "g ` {0..<1} \<subseteq> S" "pathstart g = x" "pathfinish g \<in> V"
 proof -
   have "V \<subseteq> frontier S"
@@ -2136,8 +2136,8 @@
 lemma dense_access_fp_aux:
   fixes S :: "'a::{complete_space,real_normed_vector} set"
   assumes S: "open S" "connected S"
-      and opeSU: "openin (subtopology euclidean (frontier S)) U"
-      and opeSV: "openin (subtopology euclidean (frontier S)) V"
+      and opeSU: "openin (top_of_set (frontier S)) U"
+      and opeSV: "openin (top_of_set (frontier S)) V"
       and "V \<noteq> {}" "\<not> U \<subseteq> V"
   obtains g where "arc g" "pathstart g \<in> U" "pathfinish g \<in> V" "g ` {0<..<1} \<subseteq> S"
 proof -
@@ -2150,7 +2150,7 @@
   proof (rule dense_accessible_frontier_points_connected [OF S \<open>x \<in> S\<close>])
     show "U - {pathfinish g} \<noteq> {}"
       using \<open>pathfinish g \<in> V\<close> \<open>\<not> U \<subseteq> V\<close> by blast
-    show "openin (subtopology euclidean (frontier S)) (U - {pathfinish g})"
+    show "openin (top_of_set (frontier S)) (U - {pathfinish g})"
       by (simp add: opeSU openin_delete)
   qed auto
   obtain \<gamma> where "arc \<gamma>"
@@ -2183,8 +2183,8 @@
 lemma dense_accessible_frontier_point_pairs:
   fixes S :: "'a::{complete_space,real_normed_vector} set"
   assumes S: "open S" "connected S"
-      and opeSU: "openin (subtopology euclidean (frontier S)) U"
-      and opeSV: "openin (subtopology euclidean (frontier S)) V"
+      and opeSU: "openin (top_of_set (frontier S)) U"
+      and opeSV: "openin (top_of_set (frontier S)) V"
       and "U \<noteq> {}" "V \<noteq> {}" "U \<noteq> V"
     obtains g where "arc g" "pathstart g \<in> U" "pathfinish g \<in> V" "g ` {0<..<1} \<subseteq> S"
 proof -
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Brouwer_Fixpoint.thy
--- a/src/HOL/Analysis/Brouwer_Fixpoint.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Brouwer_Fixpoint.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -85,7 +85,7 @@
 qed
 
 lemma retraction_imp_quotient_map:
-  "openin (subtopology euclidean S) (S \<inter> r -` U) \<longleftrightarrow> openin (subtopology euclidean T) U"
+  "openin (top_of_set S) (S \<inter> r -` U) \<longleftrightarrow> openin (top_of_set T) U"
   if retraction: "retraction S T r" and "U \<subseteq> T"
   using retraction apply (rule retractionE)
   apply (rule continuous_right_inverse_imp_quotient_map [where g=r])
@@ -233,12 +233,12 @@
 
 definition%important AR :: "'a::topological_space set \<Rightarrow> bool" where
 "AR S \<equiv> \<forall>U. \<forall>S'::('a * real) set.
-  S homeomorphic S' \<and> closedin (subtopology euclidean U) S' \<longrightarrow> S' retract_of U"
+  S homeomorphic S' \<and> closedin (top_of_set U) S' \<longrightarrow> S' retract_of U"
 
 definition%important ANR :: "'a::topological_space set \<Rightarrow> bool" where
 "ANR S \<equiv> \<forall>U. \<forall>S'::('a * real) set.
-  S homeomorphic S' \<and> closedin (subtopology euclidean U) S'
-  \<longrightarrow> (\<exists>T. openin (subtopology euclidean U) T \<and> S' retract_of T)"
+  S homeomorphic S' \<and> closedin (top_of_set U) S'
+  \<longrightarrow> (\<exists>T. openin (top_of_set U) T \<and> S' retract_of T)"
 
 definition%important ENR :: "'a::topological_space set \<Rightarrow> bool" where
 "ENR S \<equiv> \<exists>U. open U \<and> S retract_of U"
@@ -248,14 +248,14 @@
 lemma AR_imp_absolute_extensor:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "AR S" and contf: "continuous_on T f" and "f ` T \<subseteq> S"
-      and cloUT: "closedin (subtopology euclidean U) T"
+      and cloUT: "closedin (top_of_set U) T"
   obtains g where "continuous_on U g" "g ` U \<subseteq> S" "\<And>x. x \<in> T \<Longrightarrow> g x = f x"
 proof -
   have "aff_dim S < int (DIM('b \<times> real))"
     using aff_dim_le_DIM [of S] by simp
   then obtain C and S' :: "('b * real) set"
           where C: "convex C" "C \<noteq> {}"
-            and cloCS: "closedin (subtopology euclidean C) S'"
+            and cloCS: "closedin (top_of_set C) S'"
             and hom: "S homeomorphic S'"
     by (metis that homeomorphic_closedin_convex)
   then have "S' retract_of C"
@@ -296,7 +296,7 @@
 lemma AR_imp_absolute_retract:
   fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
   assumes "AR S" "S homeomorphic S'"
-      and clo: "closedin (subtopology euclidean U) S'"
+      and clo: "closedin (top_of_set U) S'"
     shows "S' retract_of U"
 proof -
   obtain g h where hom: "homeomorphism S S' g h"
@@ -334,12 +334,12 @@
   fixes S :: "'a::euclidean_space set"
   assumes "\<And>f :: 'a * real \<Rightarrow> 'a.
            \<And>U T. \<lbrakk>continuous_on T f;  f ` T \<subseteq> S;
-                  closedin (subtopology euclidean U) T\<rbrakk>
+                  closedin (top_of_set U) T\<rbrakk>
                  \<Longrightarrow> \<exists>g. continuous_on U g \<and> g ` U \<subseteq> S \<and> (\<forall>x \<in> T. g x = f x)"
   shows "AR S"
 proof (clarsimp simp: AR_def)
   fix U and T :: "('a * real) set"
-  assume "S homeomorphic T" and clo: "closedin (subtopology euclidean U) T"
+  assume "S homeomorphic T" and clo: "closedin (top_of_set U) T"
   then obtain g h where hom: "homeomorphism S T g h"
     by (force simp: homeomorphic_def)
   have h: "continuous_on T h" " h ` T \<subseteq> S"
@@ -369,7 +369,7 @@
   shows "AR S \<longleftrightarrow>
        (\<forall>f :: 'a * real \<Rightarrow> 'a.
         \<forall>U T. continuous_on T f \<longrightarrow> f ` T \<subseteq> S \<longrightarrow>
-               closedin (subtopology euclidean U) T \<longrightarrow>
+               closedin (top_of_set U) T \<longrightarrow>
                 (\<exists>g. continuous_on U g \<and> g ` U \<subseteq> S \<and> (\<forall>x \<in> T. g x = f x)))"
 apply (rule iffI)
  apply (metis AR_imp_absolute_extensor)
@@ -378,7 +378,7 @@
 
 lemma AR_imp_retract:
   fixes S :: "'a::euclidean_space set"
-  assumes "AR S \<and> closedin (subtopology euclidean U) S"
+  assumes "AR S \<and> closedin (top_of_set U) S"
     shows "S retract_of U"
 using AR_imp_absolute_retract assms homeomorphic_refl by blast
 
@@ -398,8 +398,8 @@
 lemma ANR_imp_absolute_neighbourhood_extensor:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "ANR S" and contf: "continuous_on T f" and "f ` T \<subseteq> S"
-      and cloUT: "closedin (subtopology euclidean U) T"
-  obtains V g where "T \<subseteq> V" "openin (subtopology euclidean U) V"
+      and cloUT: "closedin (top_of_set U) T"
+  obtains V g where "T \<subseteq> V" "openin (top_of_set U) V"
                     "continuous_on V g"
                     "g ` V \<subseteq> S" "\<And>x. x \<in> T \<Longrightarrow> g x = f x"
 proof -
@@ -407,10 +407,10 @@
     using aff_dim_le_DIM [of S] by simp
   then obtain C and S' :: "('b * real) set"
           where C: "convex C" "C \<noteq> {}"
-            and cloCS: "closedin (subtopology euclidean C) S'"
+            and cloCS: "closedin (top_of_set C) S'"
             and hom: "S homeomorphic S'"
     by (metis that homeomorphic_closedin_convex)
-  then obtain D where opD: "openin (subtopology euclidean C) D" and "S' retract_of D"
+  then obtain D where opD: "openin (top_of_set C) D" and "S' retract_of D"
     using \<open>ANR S\<close> by (auto simp: ANR_def)
   then obtain r where "S' \<subseteq> D" and contr: "continuous_on D r"
                   and "r ` D \<subseteq> S'" and rid: "\<And>x. x \<in> S' \<Longrightarrow> r x = x"
@@ -437,7 +437,7 @@
     show "T \<subseteq> U \<inter> f' -` D"
       using cloUT closedin_imp_subset \<open>S' \<subseteq> D\<close> \<open>f ` T \<subseteq> S\<close> eq homeomorphism_image1 homgh
       by fastforce
-    show ope: "openin (subtopology euclidean U) (U \<inter> f' -` D)"
+    show ope: "openin (top_of_set U) (U \<inter> f' -` D)"
       using  \<open>f' ` U \<subseteq> C\<close> by (auto simp: opD contf' continuous_openin_preimage)
     have conth: "continuous_on (r ` f' ` (U \<inter> f' -` D)) h"
       apply (rule continuous_on_subset [of S'])
@@ -460,8 +460,8 @@
 corollary ANR_imp_absolute_neighbourhood_retract:
   fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
   assumes "ANR S" "S homeomorphic S'"
-      and clo: "closedin (subtopology euclidean U) S'"
-  obtains V where "openin (subtopology euclidean U) V" "S' retract_of V"
+      and clo: "closedin (top_of_set U) S'"
+  obtains V where "openin (top_of_set U) V" "S' retract_of V"
 proof -
   obtain g h where hom: "homeomorphism S S' g h"
     using assms by (force simp: homeomorphic_def)
@@ -470,7 +470,7 @@
     apply (metis hom equalityE homeomorphism_def)
     done
     from ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> h clo]
-  obtain V h' where "S' \<subseteq> V" and opUV: "openin (subtopology euclidean U) V"
+  obtain V h' where "S' \<subseteq> V" and opUV: "openin (top_of_set U) V"
                 and h': "continuous_on V h'" "h' ` V \<subseteq> S"
                 and h'h:"\<And>x. x \<in> S' \<Longrightarrow> h' x = h x"
     by (blast intro: ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> h clo])
@@ -508,20 +508,20 @@
   fixes S :: "'a::euclidean_space set"
   assumes "\<And>f :: 'a * real \<Rightarrow> 'a.
            \<And>U T. \<lbrakk>continuous_on T f;  f ` T \<subseteq> S;
-                  closedin (subtopology euclidean U) T\<rbrakk>
-                 \<Longrightarrow> \<exists>V g. T \<subseteq> V \<and> openin (subtopology euclidean U) V \<and>
+                  closedin (top_of_set U) T\<rbrakk>
+                 \<Longrightarrow> \<exists>V g. T \<subseteq> V \<and> openin (top_of_set U) V \<and>
                        continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x \<in> T. g x = f x)"
   shows "ANR S"
 proof (clarsimp simp: ANR_def)
   fix U and T :: "('a * real) set"
-  assume "S homeomorphic T" and clo: "closedin (subtopology euclidean U) T"
+  assume "S homeomorphic T" and clo: "closedin (top_of_set U) T"
   then obtain g h where hom: "homeomorphism S T g h"
     by (force simp: homeomorphic_def)
   have h: "continuous_on T h" " h ` T \<subseteq> S"
     using hom homeomorphism_def apply blast
     apply (metis hom equalityE homeomorphism_def)
     done
-  obtain V h' where "T \<subseteq> V" and opV: "openin (subtopology euclidean U) V"
+  obtain V h' where "T \<subseteq> V" and opV: "openin (top_of_set U) V"
                 and h': "continuous_on V h'" "h' ` V \<subseteq> S"
               and h'h: "\<forall>x\<in>T. h' x = h x"
     using assms [OF h clo] by blast
@@ -538,7 +538,7 @@
     show "\<forall>x\<in>T. (g \<circ> h') x = x"
       by clarsimp (metis h'h hom homeomorphism_def)
   qed
-  then show "\<exists>V. openin (subtopology euclidean U) V \<and> T retract_of V"
+  then show "\<exists>V. openin (top_of_set U) V \<and> T retract_of V"
     using opV by blast
 qed
 
@@ -547,8 +547,8 @@
   shows "ANR S \<longleftrightarrow>
          (\<forall>f :: 'a * real \<Rightarrow> 'a.
           \<forall>U T. continuous_on T f \<longrightarrow> f ` T \<subseteq> S \<longrightarrow>
-                closedin (subtopology euclidean U) T \<longrightarrow>
-               (\<exists>V g. T \<subseteq> V \<and> openin (subtopology euclidean U) V \<and>
+                closedin (top_of_set U) T \<longrightarrow>
+               (\<exists>V g. T \<subseteq> V \<and> openin (top_of_set U) V \<and>
                        continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x \<in> T. g x = f x)))"
 apply (rule iffI)
  apply (metis ANR_imp_absolute_neighbourhood_extensor)
@@ -557,27 +557,27 @@
 
 lemma ANR_imp_neighbourhood_retract:
   fixes S :: "'a::euclidean_space set"
-  assumes "ANR S" "closedin (subtopology euclidean U) S"
-  obtains V where "openin (subtopology euclidean U) V" "S retract_of V"
+  assumes "ANR S" "closedin (top_of_set U) S"
+  obtains V where "openin (top_of_set U) V" "S retract_of V"
 using ANR_imp_absolute_neighbourhood_retract assms homeomorphic_refl by blast
 
 lemma ANR_imp_absolute_closed_neighbourhood_retract:
   fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
-  assumes "ANR S" "S homeomorphic S'" and US': "closedin (subtopology euclidean U) S'"
+  assumes "ANR S" "S homeomorphic S'" and US': "closedin (top_of_set U) S'"
   obtains V W
-    where "openin (subtopology euclidean U) V"
-          "closedin (subtopology euclidean U) W"
+    where "openin (top_of_set U) V"
+          "closedin (top_of_set U) W"
           "S' \<subseteq> V" "V \<subseteq> W" "S' retract_of W"
 proof -
-  obtain Z where "openin (subtopology euclidean U) Z" and S'Z: "S' retract_of Z"
+  obtain Z where "openin (top_of_set U) Z" and S'Z: "S' retract_of Z"
     by (blast intro: assms ANR_imp_absolute_neighbourhood_retract)
-  then have UUZ: "closedin (subtopology euclidean U) (U - Z)"
+  then have UUZ: "closedin (top_of_set U) (U - Z)"
     by auto
   have "S' \<inter> (U - Z) = {}"
     using \<open>S' retract_of Z\<close> closedin_retract closedin_subtopology by fastforce
   then obtain V W
-      where "openin (subtopology euclidean U) V"
-        and "openin (subtopology euclidean U) W"
+      where "openin (top_of_set U) V"
+        and "openin (top_of_set U) W"
         and "S' \<subseteq> V" "U - Z \<subseteq> W" "V \<inter> W = {}"
       using separation_normal_local [OF US' UUZ]  by auto
   moreover have "S' retract_of U - W"
@@ -592,9 +592,9 @@
 
 lemma ANR_imp_closed_neighbourhood_retract:
   fixes S :: "'a::euclidean_space set"
-  assumes "ANR S" "closedin (subtopology euclidean U) S"
-  obtains V W where "openin (subtopology euclidean U) V"
-                    "closedin (subtopology euclidean U) W"
+  assumes "ANR S" "closedin (top_of_set U) S"
+  obtains V W where "openin (top_of_set U) V"
+                    "closedin (top_of_set U) W"
                     "S \<subseteq> V" "V \<subseteq> W" "S retract_of W"
 by (meson ANR_imp_absolute_closed_neighbourhood_retract assms homeomorphic_refl)
 
@@ -616,7 +616,7 @@
   fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
   assumes "ENR S" and hom: "S homeomorphic S'"
       and "S' \<subseteq> U"
-  obtains V where "openin (subtopology euclidean U) V" "S' retract_of V"
+  obtains V where "openin (top_of_set U) V" "S' retract_of V"
 proof -
   obtain X where "open X" "S retract_of X"
     using \<open>ENR S\<close> by (auto simp: ENR_def)
@@ -625,8 +625,8 @@
   have "locally compact S'"
     using retract_of_locally_compact open_imp_locally_compact
           homeomorphic_local_compactness \<open>S retract_of X\<close> \<open>open X\<close> hom by blast
-  then obtain W where UW: "openin (subtopology euclidean U) W"
-                  and WS': "closedin (subtopology euclidean W) S'"
+  then obtain W where UW: "openin (top_of_set U) W"
+                  and WS': "closedin (top_of_set W) S'"
     apply (rule locally_compact_closedin_open)
     apply (rename_tac W)
     apply (rule_tac W = "U \<inter> W" in that, blast)
@@ -752,7 +752,7 @@
 proof
   assume ?lhs
   obtain C and S' :: "('a * real) set"
-    where "convex C" "C \<noteq> {}" "closedin (subtopology euclidean C) S'" "S homeomorphic S'"
+    where "convex C" "C \<noteq> {}" "closedin (top_of_set C) S'" "S homeomorphic S'"
       apply (rule homeomorphic_closedin_convex [of S, where 'n = "'a * real"])
       using aff_dim_le_DIM [of S] by auto
   with \<open>AR S\<close> have "contractible S"
@@ -777,27 +777,27 @@
     by (metis all_not_in_conv atLeastAtMost_iff image_subset_iff mem_Sigma_iff order_refl zero_le_one)
   have "\<exists>g. continuous_on W g \<and> g ` W \<subseteq> S \<and> (\<forall>x\<in>T. g x = f x)"
          if      f: "continuous_on T f" "f ` T \<subseteq> S"
-            and WT: "closedin (subtopology euclidean W) T"
+            and WT: "closedin (top_of_set W) T"
          for W T and f :: "'a \<times> real \<Rightarrow> 'a"
   proof -
     obtain U g
-      where "T \<subseteq> U" and WU: "openin (subtopology euclidean W) U"
+      where "T \<subseteq> U" and WU: "openin (top_of_set W) U"
         and contg: "continuous_on U g"
         and "g ` U \<subseteq> S" and gf: "\<And>x. x \<in> T \<Longrightarrow> g x = f x"
       using iffD1 [OF ANR_eq_absolute_neighbourhood_extensor \<open>ANR S\<close>, rule_format, OF f WT]
       by auto
-    have WWU: "closedin (subtopology euclidean W) (W - U)"
+    have WWU: "closedin (top_of_set W) (W - U)"
       using WU closedin_diff by fastforce
     moreover have "(W - U) \<inter> T = {}"
       using \<open>T \<subseteq> U\<close> by auto
     ultimately obtain V V'
-      where WV': "openin (subtopology euclidean W) V'"
-        and WV: "openin (subtopology euclidean W) V"
+      where WV': "openin (top_of_set W) V'"
+        and WV: "openin (top_of_set W) V"
         and "W - U \<subseteq> V'" "T \<subseteq> V" "V' \<inter> V = {}"
       using separation_normal_local [of W "W-U" T] WT by blast
     then have WVT: "T \<inter> (W - V) = {}"
       by auto
-    have WWV: "closedin (subtopology euclidean W) (W - V)"
+    have WWV: "closedin (top_of_set W) (W - V)"
       using WV closedin_diff by fastforce
     obtain j :: " 'a \<times> real \<Rightarrow> real"
       where contj: "continuous_on W j"
@@ -931,8 +931,8 @@
 
 lemma AR_closed_Un_local_aux:
   fixes U :: "'a::euclidean_space set"
-  assumes "closedin (subtopology euclidean U) S"
-          "closedin (subtopology euclidean U) T"
+  assumes "closedin (top_of_set U) S"
+          "closedin (top_of_set U) T"
           "AR S" "AR T" "AR(S \<inter> T)"
   shows "(S \<union> T) retract_of U"
 proof -
@@ -943,10 +943,10 @@
   define S' where "S' \<equiv> {x \<in> U. setdist {x} S \<le> setdist {x} T}"
   define T' where "T' \<equiv> {x \<in> U. setdist {x} T \<le> setdist {x} S}"
   define W  where "W \<equiv> {x \<in> U. setdist {x} S = setdist {x} T}"
-  have US': "closedin (subtopology euclidean U) S'"
+  have US': "closedin (top_of_set U) S'"
     using continuous_closedin_preimage [of U "\<lambda>x. setdist {x} S - setdist {x} T" "{..0}"]
     by (simp add: S'_def vimage_def Collect_conj_eq continuous_on_diff continuous_on_setdist)
-  have UT': "closedin (subtopology euclidean U) T'"
+  have UT': "closedin (top_of_set U) T'"
     using continuous_closedin_preimage [of U "\<lambda>x. setdist {x} T - setdist {x} S" "{..0}"]
     by (simp add: T'_def vimage_def Collect_conj_eq continuous_on_diff continuous_on_setdist)
   have "S \<subseteq> S'"
@@ -971,7 +971,7 @@
     by (force simp: W_def setdist_sing_in_set)
   have "S' \<inter> T' = W"
     by (auto simp: S'_def T'_def W_def)
-  then have cloUW: "closedin (subtopology euclidean U) W"
+  then have cloUW: "closedin (top_of_set U) W"
     using closedin_Int US' UT' by blast
   define r where "r \<equiv> \<lambda>x. if x \<in> W then r0 x else x"
   have "r ` (W \<union> S) \<subseteq> S" "r ` (W \<union> T) \<subseteq> T"
@@ -979,14 +979,14 @@
   have contr: "continuous_on (W \<union> (S \<union> T)) r"
   unfolding r_def
   proof (rule continuous_on_cases_local [OF _ _ contr0 continuous_on_id])
-    show "closedin (subtopology euclidean (W \<union> (S \<union> T))) W"
-      using \<open>S \<subseteq> U\<close> \<open>T \<subseteq> U\<close> \<open>W \<subseteq> U\<close> \<open>closedin (subtopology euclidean U) W\<close> closedin_subset_trans by fastforce
-    show "closedin (subtopology euclidean (W \<union> (S \<union> T))) (S \<union> T)"
+    show "closedin (top_of_set (W \<union> (S \<union> T))) W"
+      using \<open>S \<subseteq> U\<close> \<open>T \<subseteq> U\<close> \<open>W \<subseteq> U\<close> \<open>closedin (top_of_set U) W\<close> closedin_subset_trans by fastforce
+    show "closedin (top_of_set (W \<union> (S \<union> T))) (S \<union> T)"
       by (meson \<open>S \<subseteq> U\<close> \<open>T \<subseteq> U\<close> \<open>W \<subseteq> U\<close> assms closedin_Un closedin_subset_trans sup.bounded_iff sup.cobounded2)
     show "\<And>x. x \<in> W \<and> x \<notin> W \<or> x \<in> S \<union> T \<and> x \<in> W \<Longrightarrow> r0 x = x"
       by (auto simp: ST)
   qed
-  have cloUWS: "closedin (subtopology euclidean U) (W \<union> S)"
+  have cloUWS: "closedin (top_of_set U) (W \<union> S)"
     by (simp add: cloUW assms closedin_Un)
   obtain g where contg: "continuous_on U g"
              and "g ` U \<subseteq> S" and geqr: "\<And>x. x \<in> W \<union> S \<Longrightarrow> g x = r x"
@@ -994,7 +994,7 @@
       apply (rule continuous_on_subset [OF contr])
       using \<open>r ` (W \<union> S) \<subseteq> S\<close> apply auto
     done
-  have cloUWT: "closedin (subtopology euclidean U) (W \<union> T)"
+  have cloUWT: "closedin (top_of_set U) (W \<union> T)"
     by (simp add: cloUW assms closedin_Un)
   obtain h where conth: "continuous_on U h"
              and "h ` U \<subseteq> T" and heqr: "\<And>x. x \<in> W \<union> T \<Longrightarrow> h x = r x"
@@ -1022,24 +1022,24 @@
 
 lemma AR_closed_Un_local:
   fixes S :: "'a::euclidean_space set"
-  assumes STS: "closedin (subtopology euclidean (S \<union> T)) S"
-      and STT: "closedin (subtopology euclidean (S \<union> T)) T"
+  assumes STS: "closedin (top_of_set (S \<union> T)) S"
+      and STT: "closedin (top_of_set (S \<union> T)) T"
       and "AR S" "AR T" "AR(S \<inter> T)"
     shows "AR(S \<union> T)"
 proof -
   have "C retract_of U"
-       if hom: "S \<union> T homeomorphic C" and UC: "closedin (subtopology euclidean U) C"
+       if hom: "S \<union> T homeomorphic C" and UC: "closedin (top_of_set U) C"
        for U and C :: "('a * real) set"
   proof -
     obtain f g where hom: "homeomorphism (S \<union> T) C f g"
       using hom by (force simp: homeomorphic_def)
-    have US: "closedin (subtopology euclidean U) (C \<inter> g -` S)"
+    have US: "closedin (top_of_set U) (C \<inter> g -` S)"
       apply (rule closedin_trans [OF _ UC])
       apply (rule continuous_closedin_preimage_gen [OF _ _ STS])
       using hom homeomorphism_def apply blast
       apply (metis hom homeomorphism_def set_eq_subset)
       done
-    have UT: "closedin (subtopology euclidean U) (C \<inter> g -` T)"
+    have UT: "closedin (top_of_set U) (C \<inter> g -` T)"
       apply (rule closedin_trans [OF _ UC])
       apply (rule continuous_closedin_preimage_gen [OF _ _ STT])
       using hom homeomorphism_def apply blast
@@ -1088,10 +1088,10 @@
 
 lemma ANR_closed_Un_local_aux:
   fixes U :: "'a::euclidean_space set"
-  assumes US: "closedin (subtopology euclidean U) S"
-      and UT: "closedin (subtopology euclidean U) T"
+  assumes US: "closedin (top_of_set U) S"
+      and UT: "closedin (top_of_set U) T"
       and "ANR S" "ANR T" "ANR(S \<inter> T)"
-  obtains V where "openin (subtopology euclidean U) V" "(S \<union> T) retract_of V"
+  obtains V where "openin (top_of_set U) V" "(S \<union> T) retract_of V"
 proof (cases "S = {} \<or> T = {}")
   case True with assms that show ?thesis
     by (metis ANR_imp_neighbourhood_retract Un_commute inf_bot_right sup_inf_absorb)
@@ -1103,10 +1103,10 @@
   define S' where "S' \<equiv> {x \<in> U. setdist {x} S \<le> setdist {x} T}"
   define T' where "T' \<equiv> {x \<in> U. setdist {x} T \<le> setdist {x} S}"
   define W  where "W \<equiv> {x \<in> U. setdist {x} S = setdist {x} T}"
-  have cloUS': "closedin (subtopology euclidean U) S'"
+  have cloUS': "closedin (top_of_set U) S'"
     using continuous_closedin_preimage [of U "\<lambda>x. setdist {x} S - setdist {x} T" "{..0}"]
     by (simp add: S'_def vimage_def Collect_conj_eq continuous_on_diff continuous_on_setdist)
-  have cloUT': "closedin (subtopology euclidean U) T'"
+  have cloUT': "closedin (top_of_set U) T'"
     using continuous_closedin_preimage [of U "\<lambda>x. setdist {x} T - setdist {x} S" "{..0}"]
     by (simp add: T'_def vimage_def Collect_conj_eq continuous_on_diff continuous_on_setdist)
   have "S \<subseteq> S'"
@@ -1123,17 +1123,17 @@
     using \<open>S \<subseteq> U\<close> by (force simp: W_def setdist_sing_in_set)+
   have "S' \<inter> T' = W"
     by (auto simp: S'_def T'_def W_def)
-  then have cloUW: "closedin (subtopology euclidean U) W"
+  then have cloUW: "closedin (top_of_set U) W"
     using closedin_Int cloUS' cloUT' by blast
-  obtain W' W0 where "openin (subtopology euclidean W) W'"
-                 and cloWW0: "closedin (subtopology euclidean W) W0"
+  obtain W' W0 where "openin (top_of_set W) W'"
+                 and cloWW0: "closedin (top_of_set W) W0"
                  and "S \<inter> T \<subseteq> W'" "W' \<subseteq> W0"
                  and ret: "(S \<inter> T) retract_of W0"
     apply (rule ANR_imp_closed_neighbourhood_retract [OF \<open>ANR(S \<inter> T)\<close>])
     apply (rule closedin_subset_trans [of U, OF _ ST_W \<open>W \<subseteq> U\<close>])
     apply (blast intro: assms)+
     done
-  then obtain U0 where opeUU0: "openin (subtopology euclidean U) U0"
+  then obtain U0 where opeUU0: "openin (top_of_set U) U0"
                    and U0: "S \<inter> T \<subseteq> U0" "U0 \<inter> W \<subseteq> W0"
     unfolding openin_open  using \<open>W \<subseteq> U\<close> by blast
   have "W0 \<subseteq> U"
@@ -1150,29 +1150,29 @@
   have contr: "continuous_on (W0 \<union> (S \<union> T)) r"
   unfolding r_def
   proof (rule continuous_on_cases_local [OF _ _ contr0 continuous_on_id])
-    show "closedin (subtopology euclidean (W0 \<union> (S \<union> T))) W0"
+    show "closedin (top_of_set (W0 \<union> (S \<union> T))) W0"
       apply (rule closedin_subset_trans [of U])
       using cloWW0 cloUW closedin_trans \<open>W0 \<subseteq> U\<close> \<open>S \<subseteq> U\<close> \<open>T \<subseteq> U\<close> apply blast+
       done
-    show "closedin (subtopology euclidean (W0 \<union> (S \<union> T))) (S \<union> T)"
+    show "closedin (top_of_set (W0 \<union> (S \<union> T))) (S \<union> T)"
       by (meson \<open>S \<subseteq> U\<close> \<open>T \<subseteq> U\<close> \<open>W0 \<subseteq> U\<close> assms closedin_Un closedin_subset_trans sup.bounded_iff sup.cobounded2)
     show "\<And>x. x \<in> W0 \<and> x \<notin> W0 \<or> x \<in> S \<union> T \<and> x \<in> W0 \<Longrightarrow> r0 x = x"
       using ST cloWW0 closedin_subset by fastforce
   qed
-  have cloS'WS: "closedin (subtopology euclidean S') (W0 \<union> S)"
+  have cloS'WS: "closedin (top_of_set S') (W0 \<union> S)"
     by (meson closedin_subset_trans US cloUS' \<open>S \<subseteq> S'\<close> \<open>W \<subseteq> S'\<close> cloUW cloWW0 
               closedin_Un closedin_imp_subset closedin_trans)
   obtain W1 g where "W0 \<union> S \<subseteq> W1" and contg: "continuous_on W1 g"
-                and opeSW1: "openin (subtopology euclidean S') W1"
+                and opeSW1: "openin (top_of_set S') W1"
                 and "g ` W1 \<subseteq> S" and geqr: "\<And>x. x \<in> W0 \<union> S \<Longrightarrow> g x = r x"
     apply (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> _ \<open>r ` (W0 \<union> S) \<subseteq> S\<close> cloS'WS])
      apply (rule continuous_on_subset [OF contr], blast+)
     done
-  have cloT'WT: "closedin (subtopology euclidean T') (W0 \<union> T)"
+  have cloT'WT: "closedin (top_of_set T') (W0 \<union> T)"
     by (meson closedin_subset_trans UT cloUT' \<open>T \<subseteq> T'\<close> \<open>W \<subseteq> T'\<close> cloUW cloWW0 
               closedin_Un closedin_imp_subset closedin_trans)
   obtain W2 h where "W0 \<union> T \<subseteq> W2" and conth: "continuous_on W2 h"
-                and opeSW2: "openin (subtopology euclidean T') W2"
+                and opeSW2: "openin (top_of_set T') W2"
                 and "h ` W2 \<subseteq> T" and heqr: "\<And>x. x \<in> W0 \<union> T \<Longrightarrow> h x = r x"
     apply (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> _ \<open>r ` (W0 \<union> T) \<subseteq> T\<close> cloT'WT])
      apply (rule continuous_on_subset [OF contr], blast+)
@@ -1187,17 +1187,17 @@
          ((U - T') \<inter> O1 \<union> (U - S') \<inter> O2 \<union> U \<inter> O1 \<inter> O2) - (W - U0)"
      using \<open>U0 \<inter> W \<subseteq> W0\<close> \<open>W0 \<union> S \<subseteq> W1\<close> \<open>W0 \<union> T \<subseteq> W2\<close>
       by (auto simp: \<open>S' \<union> T' = U\<close> [symmetric] \<open>S' \<inter> T' = W\<close> [symmetric] \<open>W1 = S' \<inter> O1\<close> \<open>W2 = T' \<inter> O2\<close>)
-    show "openin (subtopology euclidean U) (W1 - (W - U0) \<union> (W2 - (W - U0)))"
+    show "openin (top_of_set U) (W1 - (W - U0) \<union> (W2 - (W - U0)))"
       apply (subst eq)
       apply (intro openin_Un openin_Int_open openin_diff closedin_diff cloUW opeUU0 cloUS' cloUT' \<open>open O1\<close> \<open>open O2\<close>, simp_all)
       done
-    have cloW1: "closedin (subtopology euclidean (W1 - (W - U0) \<union> (W2 - (W - U0)))) (W1 - (W - U0))"
+    have cloW1: "closedin (top_of_set (W1 - (W - U0) \<union> (W2 - (W - U0)))) (W1 - (W - U0))"
       using cloUS' apply (simp add: closedin_closed)
       apply (erule ex_forward)
       using U0 \<open>W0 \<union> S \<subseteq> W1\<close>
       apply (auto simp: \<open>W1 = S' \<inter> O1\<close> \<open>W2 = T' \<inter> O2\<close> \<open>S' \<union> T' = U\<close> [symmetric]\<open>S' \<inter> T' = W\<close> [symmetric])
       done
-    have cloW2: "closedin (subtopology euclidean (W1 - (W - U0) \<union> (W2 - (W - U0)))) (W2 - (W - U0))"
+    have cloW2: "closedin (top_of_set (W1 - (W - U0) \<union> (W2 - (W - U0)))) (W2 - (W - U0))"
       using cloUT' apply (simp add: closedin_closed)
       apply (erule ex_forward)
       using U0 \<open>W0 \<union> T \<subseteq> W2\<close>
@@ -1227,24 +1227,24 @@
 
 lemma ANR_closed_Un_local:
   fixes S :: "'a::euclidean_space set"
-  assumes STS: "closedin (subtopology euclidean (S \<union> T)) S"
-      and STT: "closedin (subtopology euclidean (S \<union> T)) T"
+  assumes STS: "closedin (top_of_set (S \<union> T)) S"
+      and STT: "closedin (top_of_set (S \<union> T)) T"
       and "ANR S" "ANR T" "ANR(S \<inter> T)" 
     shows "ANR(S \<union> T)"
 proof -
-  have "\<exists>T. openin (subtopology euclidean U) T \<and> C retract_of T"
-       if hom: "S \<union> T homeomorphic C" and UC: "closedin (subtopology euclidean U) C"
+  have "\<exists>T. openin (top_of_set U) T \<and> C retract_of T"
+       if hom: "S \<union> T homeomorphic C" and UC: "closedin (top_of_set U) C"
        for U and C :: "('a * real) set"
   proof -
     obtain f g where hom: "homeomorphism (S \<union> T) C f g"
       using hom by (force simp: homeomorphic_def)
-    have US: "closedin (subtopology euclidean U) (C \<inter> g -` S)"
+    have US: "closedin (top_of_set U) (C \<inter> g -` S)"
       apply (rule closedin_trans [OF _ UC])
       apply (rule continuous_closedin_preimage_gen [OF _ _ STS])
       using hom [unfolded homeomorphism_def] apply blast
       apply (metis hom homeomorphism_def set_eq_subset)
       done
-    have UT: "closedin (subtopology euclidean U) (C \<inter> g -` T)"
+    have UT: "closedin (top_of_set U) (C \<inter> g -` T)"
       apply (rule closedin_trans [OF _ UC])
       apply (rule continuous_closedin_preimage_gen [OF _ _ STT])
       using hom [unfolded homeomorphism_def] apply blast
@@ -1291,26 +1291,26 @@
 
 lemma ANR_openin:
   fixes S :: "'a::euclidean_space set"
-  assumes "ANR T" and opeTS: "openin (subtopology euclidean T) S"
+  assumes "ANR T" and opeTS: "openin (top_of_set T) S"
   shows "ANR S"
 proof (clarsimp simp only: ANR_eq_absolute_neighbourhood_extensor)
   fix f :: "'a \<times> real \<Rightarrow> 'a" and U C
   assume contf: "continuous_on C f" and fim: "f ` C \<subseteq> S"
-     and cloUC: "closedin (subtopology euclidean U) C"
+     and cloUC: "closedin (top_of_set U) C"
   have "f ` C \<subseteq> T"
     using fim opeTS openin_imp_subset by blast
   obtain W g where "C \<subseteq> W"
-               and UW: "openin (subtopology euclidean U) W"
+               and UW: "openin (top_of_set U) W"
                and contg: "continuous_on W g"
                and gim: "g ` W \<subseteq> T"
                and geq: "\<And>x. x \<in> C \<Longrightarrow> g x = f x"
     apply (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> contf \<open>f ` C \<subseteq> T\<close> cloUC])
     using fim by auto
-  show "\<exists>V g. C \<subseteq> V \<and> openin (subtopology euclidean U) V \<and> continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x\<in>C. g x = f x)"
+  show "\<exists>V g. C \<subseteq> V \<and> openin (top_of_set U) V \<and> continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x\<in>C. g x = f x)"
   proof (intro exI conjI)
     show "C \<subseteq> W \<inter> g -` S"
       using \<open>C \<subseteq> W\<close> fim geq by blast
-    show "openin (subtopology euclidean U) (W \<inter> g -` S)"
+    show "openin (top_of_set U) (W \<inter> g -` S)"
       by (metis (mono_tags, lifting) UW contg continuous_openin_preimage gim opeTS openin_trans)
     show "continuous_on (W \<inter> g -` S) g"
       by (blast intro: continuous_on_subset [OF contg])
@@ -1323,20 +1323,20 @@
 
 lemma ENR_openin:
     fixes S :: "'a::euclidean_space set"
-    assumes "ENR T" and opeTS: "openin (subtopology euclidean T) S"
+    assumes "ENR T" and opeTS: "openin (top_of_set T) S"
     shows "ENR S"
   using assms apply (simp add: ENR_ANR)
   using ANR_openin locally_open_subset by blast
 
 lemma ANR_neighborhood_retract:
     fixes S :: "'a::euclidean_space set"
-    assumes "ANR U" "S retract_of T" "openin (subtopology euclidean U) T"
+    assumes "ANR U" "S retract_of T" "openin (top_of_set U) T"
     shows "ANR S"
   using ANR_openin ANR_retract_of_ANR assms by blast
 
 lemma ENR_neighborhood_retract:
     fixes S :: "'a::euclidean_space set"
-    assumes "ENR U" "S retract_of T" "openin (subtopology euclidean U) T"
+    assumes "ENR U" "S retract_of T" "openin (top_of_set U) T"
     shows "ENR S"
   using ENR_openin ENR_retract_of_ENR assms by blast
 
@@ -1438,7 +1438,7 @@
 proof -
   obtain U and T :: "('a \<times> real) set"
      where "convex U" "U \<noteq> {}"
-       and UT: "closedin (subtopology euclidean U) T"
+       and UT: "closedin (top_of_set U) T"
        and "S homeomorphic T"
     apply (rule homeomorphic_closedin_convex [of S])
     using aff_dim_le_DIM [of S] apply auto
@@ -1447,7 +1447,7 @@
     by (meson ANR_imp_absolute_neighbourhood_retract
         assms convex_imp_locally_path_connected locally_open_subset retract_of_locally_path_connected)
   then have S: "locally path_connected S"
-      if "openin (subtopology euclidean U) V" "T retract_of V" "U \<noteq> {}" for V
+      if "openin (top_of_set U) V" "T retract_of V" "U \<noteq> {}" for V
     using \<open>S homeomorphic T\<close> homeomorphic_locally homeomorphic_path_connectedness by blast
   show ?thesis
     using assms
@@ -1494,11 +1494,11 @@
 proof (clarsimp simp only: ANR_eq_absolute_neighbourhood_extensor)
   fix f :: " ('a \<times> 'b) \<times> real \<Rightarrow> 'a \<times> 'b" and U C
   assume "continuous_on C f" and fim: "f ` C \<subseteq> S \<times> T"
-     and cloUC: "closedin (subtopology euclidean U) C"
+     and cloUC: "closedin (top_of_set U) C"
   have contf1: "continuous_on C (fst \<circ> f)"
     by (simp add: \<open>continuous_on C f\<close> continuous_on_fst)
   obtain W1 g where "C \<subseteq> W1"
-               and UW1: "openin (subtopology euclidean U) W1"
+               and UW1: "openin (top_of_set U) W1"
                and contg: "continuous_on W1 g"
                and gim: "g ` W1 \<subseteq> S"
                and geq: "\<And>x. x \<in> C \<Longrightarrow> g x = (fst \<circ> f) x"
@@ -1508,7 +1508,7 @@
   have contf2: "continuous_on C (snd \<circ> f)"
     by (simp add: \<open>continuous_on C f\<close> continuous_on_snd)
   obtain W2 h where "C \<subseteq> W2"
-               and UW2: "openin (subtopology euclidean U) W2"
+               and UW2: "openin (top_of_set U) W2"
                and conth: "continuous_on W2 h"
                and him: "h ` W2 \<subseteq> T"
                and heq: "\<And>x. x \<in> C \<Longrightarrow> h x = (snd \<circ> f) x"
@@ -1516,12 +1516,12 @@
     using fim apply auto
     done
   show "\<exists>V g. C \<subseteq> V \<and>
-               openin (subtopology euclidean U) V \<and>
+               openin (top_of_set U) V \<and>
                continuous_on V g \<and> g ` V \<subseteq> S \<times> T \<and> (\<forall>x\<in>C. g x = f x)"
   proof (intro exI conjI)
     show "C \<subseteq> W1 \<inter> W2"
       by (simp add: \<open>C \<subseteq> W1\<close> \<open>C \<subseteq> W2\<close>)
-    show "openin (subtopology euclidean U) (W1 \<inter> W2)"
+    show "openin (top_of_set U) (W1 \<inter> W2)"
       by (simp add: UW1 UW2 openin_Int)
     show  "continuous_on (W1 \<inter> W2) (\<lambda>x. (g x, h x))"
       by (metis (no_types) contg conth continuous_on_Pair continuous_on_subset inf_commute inf_le1)
@@ -3706,7 +3706,7 @@
     apply (auto simp: False closest_point_self affine_imp_convex closest_point_in_set continuous_on_closest_point)
     done
   finally have "rel_frontier S retract_of {x. closest_point (affine hull S) x \<noteq> a}" .
-  moreover have "openin (subtopology euclidean UNIV) (UNIV \<inter> closest_point (affine hull S) -` (- {a}))"
+  moreover have "openin (top_of_set UNIV) (UNIV \<inter> closest_point (affine hull S) -` (- {a}))"
     apply (rule continuous_openin_preimage_gen)
     apply (auto simp: False affine_imp_convex continuous_on_closest_point)
     done
@@ -3726,7 +3726,7 @@
 lemma ENR_closedin_Un_local:
   fixes S :: "'a::euclidean_space set"
   shows "\<lbrakk>ENR S; ENR T; ENR(S \<inter> T);
-          closedin (subtopology euclidean (S \<union> T)) S; closedin (subtopology euclidean (S \<union> T)) T\<rbrakk>
+          closedin (top_of_set (S \<union> T)) S; closedin (top_of_set (S \<union> T)) T\<rbrakk>
         \<Longrightarrow> ENR(S \<union> T)"
 by (simp add: ENR_ANR ANR_closed_Un_local locally_compact_closedin_Un)
 
@@ -3743,8 +3743,8 @@
 
 lemma retract_from_Un_Int:
   fixes S :: "'a::euclidean_space set"
-  assumes clS: "closedin (subtopology euclidean (S \<union> T)) S"
-      and clT: "closedin (subtopology euclidean (S \<union> T)) T"
+  assumes clS: "closedin (top_of_set (S \<union> T)) S"
+      and clT: "closedin (top_of_set (S \<union> T)) T"
       and Un: "(S \<union> T) retract_of U" and Int: "(S \<inter> T) retract_of T"
     shows "S retract_of U"
 proof -
@@ -3764,8 +3764,8 @@
 
 lemma AR_from_Un_Int_local:
   fixes S :: "'a::euclidean_space set"
-  assumes clS: "closedin (subtopology euclidean (S \<union> T)) S"
-      and clT: "closedin (subtopology euclidean (S \<union> T)) T"
+  assumes clS: "closedin (top_of_set (S \<union> T)) S"
+      and clT: "closedin (top_of_set (S \<union> T)) T"
       and Un: "AR(S \<union> T)" and Int: "AR(S \<inter> T)"
     shows "AR S"
   apply (rule AR_retract_of_AR [OF Un])
@@ -3773,8 +3773,8 @@
 
 lemma AR_from_Un_Int_local':
   fixes S :: "'a::euclidean_space set"
-  assumes "closedin (subtopology euclidean (S \<union> T)) S"
-      and "closedin (subtopology euclidean (S \<union> T)) T"
+  assumes "closedin (top_of_set (S \<union> T)) S"
+      and "closedin (top_of_set (S \<union> T)) T"
       and "AR(S \<union> T)" "AR(S \<inter> T)"
     shows "AR T"
   using AR_from_Un_Int_local [of T S] assms by (simp add: Un_commute Int_commute)
@@ -3787,13 +3787,13 @@
 
 lemma ANR_from_Un_Int_local:
   fixes S :: "'a::euclidean_space set"
-  assumes clS: "closedin (subtopology euclidean (S \<union> T)) S"
-      and clT: "closedin (subtopology euclidean (S \<union> T)) T"
+  assumes clS: "closedin (top_of_set (S \<union> T)) S"
+      and clT: "closedin (top_of_set (S \<union> T)) T"
       and Un: "ANR(S \<union> T)" and Int: "ANR(S \<inter> T)"
     shows "ANR S"
 proof -
-  obtain V where clo: "closedin (subtopology euclidean (S \<union> T)) (S \<inter> T)"
-             and ope: "openin (subtopology euclidean (S \<union> T)) V"
+  obtain V where clo: "closedin (top_of_set (S \<union> T)) (S \<inter> T)"
+             and ope: "openin (top_of_set (S \<union> T)) V"
              and ret: "S \<inter> T retract_of V"
     using ANR_imp_neighbourhood_retract [OF Int] by (metis clS clT closedin_Int)
   then obtain r where r: "continuous_on V r" and rim: "r ` V \<subseteq> S \<inter> T" and req: "\<forall>x\<in>S \<inter> T. r x = x"
@@ -3808,9 +3808,9 @@
     using Vsub by blast
   have "continuous_on (S \<union> V \<inter> T) (\<lambda>x. if x \<in> S then x else r x)"
   proof (rule continuous_on_cases_local)
-    show "closedin (subtopology euclidean (S \<union> V \<inter> T)) S"
+    show "closedin (top_of_set (S \<union> V \<inter> T)) S"
       using clS closedin_subset_trans inf.boundedE by blast
-    show "closedin (subtopology euclidean (S \<union> V \<inter> T)) (V \<inter> T)"
+    show "closedin (top_of_set (S \<union> V \<inter> T)) (V \<inter> T)"
       using clT Vsup by (auto simp: closedin_closed)
     show "continuous_on (V \<inter> T) r"
       by (meson Int_lower1 continuous_on_subset r)
@@ -3938,7 +3938,7 @@
 
 lemma absolute_retract_imp_AR_gen:
   fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
-  assumes "S retract_of T" "convex T" "T \<noteq> {}" "S homeomorphic S'" "closedin (subtopology euclidean U) S'"
+  assumes "S retract_of T" "convex T" "T \<noteq> {}" "S homeomorphic S'" "closedin (top_of_set U) S'"
   shows "S' retract_of U"
 proof -
   have "AR T"
@@ -3972,7 +3972,7 @@
 
 lemma ENR_from_Un_Int_gen:
   fixes S :: "'a::euclidean_space set"
-  assumes "closedin (subtopology euclidean (S \<union> T)) S" "closedin (subtopology euclidean (S \<union> T)) T" "ENR(S \<union> T)" "ENR(S \<inter> T)"
+  assumes "closedin (top_of_set (S \<union> T)) S" "closedin (top_of_set (S \<union> T)) T" "ENR(S \<union> T)" "ENR(S \<inter> T)"
   shows "ENR S"
   apply (simp add: ENR_ANR)
   using ANR_from_Un_Int_local ENR_ANR assms locally_compact_closedin by blast
@@ -4124,7 +4124,7 @@
 
 theorem Borsuk_homotopy_extension_homotopic:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes cloTS: "closedin (subtopology euclidean T) S"
+  assumes cloTS: "closedin (top_of_set T) S"
       and anr: "(ANR S \<and> ANR T) \<or> ANR U"
       and contf: "continuous_on T f"
       and "f ` T \<subseteq> U"
@@ -4140,15 +4140,15 @@
        using assms by (auto simp: homotopic_with_def)
   define h' where "h' \<equiv>  \<lambda>z. if snd z \<in> S then h z else (f \<circ> snd) z"
   define B where "B \<equiv> {0::real} \<times> T \<union> {0..1} \<times> S"
-  have clo0T: "closedin (subtopology euclidean ({0..1} \<times> T)) ({0::real} \<times> T)"
+  have clo0T: "closedin (top_of_set ({0..1} \<times> T)) ({0::real} \<times> T)"
     by (simp add: closedin_subtopology_refl closedin_Times)
-  moreover have cloT1S: "closedin (subtopology euclidean ({0..1} \<times> T)) ({0..1} \<times> S)"
+  moreover have cloT1S: "closedin (top_of_set ({0..1} \<times> T)) ({0..1} \<times> S)"
     by (simp add: closedin_subtopology_refl closedin_Times assms)
-  ultimately have clo0TB:"closedin (subtopology euclidean ({0..1} \<times> T)) B"
+  ultimately have clo0TB:"closedin (top_of_set ({0..1} \<times> T)) B"
     by (auto simp: B_def)
-  have cloBS: "closedin (subtopology euclidean B) ({0..1} \<times> S)"
+  have cloBS: "closedin (top_of_set B) ({0..1} \<times> S)"
     by (metis (no_types) Un_subset_iff B_def closedin_subset_trans [OF cloT1S] clo0TB closedin_imp_subset closedin_self)
-  moreover have cloBT: "closedin (subtopology euclidean B) ({0} \<times> T)"
+  moreover have cloBT: "closedin (top_of_set B) ({0} \<times> T)"
     using \<open>S \<subseteq> T\<close> closedin_subset_trans [OF clo0T]
     by (metis B_def Un_upper1 clo0TB closedin_closed inf_le1)
   moreover have "continuous_on ({0} \<times> T) (f \<circ> snd)"
@@ -4161,7 +4161,7 @@
     done
   have "image h' B \<subseteq> U"
     using \<open>f ` T \<subseteq> U\<close> him by (auto simp: h'_def B_def)
-  obtain V k where "B \<subseteq> V" and opeTV: "openin (subtopology euclidean ({0..1} \<times> T)) V"
+  obtain V k where "B \<subseteq> V" and opeTV: "openin (top_of_set ({0..1} \<times> T)) V"
                and contk: "continuous_on V k" and kim: "k ` V \<subseteq> U"
                and keq: "\<And>x. x \<in> B \<Longrightarrow> k x = h' x"
   using anr
@@ -4177,7 +4177,7 @@
         apply (simp_all add: ANR_Times convex_imp_ANR ANR_singleton ST eq)
       done
     note Vk = that
-    have *: thesis if "openin (subtopology euclidean ({0..1::real} \<times> T)) V"
+    have *: thesis if "openin (top_of_set ({0..1::real} \<times> T)) V"
                       "retraction V B r" for V r
       using that
       apply (clarsimp simp add: retraction_def)
@@ -4195,14 +4195,14 @@
     show ?thesis by blast
   qed
   define S' where "S' \<equiv> {x. \<exists>u::real. u \<in> {0..1} \<and> (u, x::'a) \<in> {0..1} \<times> T - V}"
-  have "closedin (subtopology euclidean T) S'"
+  have "closedin (top_of_set T) S'"
     unfolding S'_def
     apply (rule closedin_compact_projection, blast)
     using closedin_self opeTV by blast
   have S'_def: "S' = {x. \<exists>u::real.  (u, x::'a) \<in> {0..1} \<times> T - V}"
     by (auto simp: S'_def)
-  have cloTS': "closedin (subtopology euclidean T) S'"
-    using S'_def \<open>closedin (subtopology euclidean T) S'\<close> by blast
+  have cloTS': "closedin (top_of_set T) S'"
+    using S'_def \<open>closedin (top_of_set T) S'\<close> by blast
   have "S \<inter> S' = {}"
     using S'_def B_def \<open>B \<subseteq> V\<close> by force
   obtain a :: "'a \<Rightarrow> real" where conta: "continuous_on T a"
@@ -4269,7 +4269,7 @@
   assume ?lhs
   then obtain c where c: "homotopic_with (\<lambda>x. True) S T (\<lambda>x. c) f"
     by (blast intro: homotopic_with_symD)
-  have "closedin (subtopology euclidean UNIV) S"
+  have "closedin (top_of_set UNIV) S"
     using \<open>closed S\<close> closed_closedin by fastforce
   then obtain g where "continuous_on UNIV g" "range g \<subseteq> T"
                       "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
@@ -4412,7 +4412,7 @@
                           "g ` (S \<union> connected_component_set (- S) a) \<subseteq> sphere 0 1"
                           "\<And>x. x \<in> S \<Longrightarrow> g x = (x - a) /\<^sub>R norm (x - a)"
   proof (rule Borsuk_homotopy_extension_homotopic)
-    show "closedin (subtopology euclidean ?T) S"
+    show "closedin (top_of_set ?T) S"
       by (simp add: \<open>compact S\<close> closed_subset compact_imp_closed)
     show "continuous_on ?T (\<lambda>x. (x - b) /\<^sub>R norm (x - b))"
       by (simp add: \<open>b \<notin> S\<close> notcc continuous_on_Borsuk_map)
@@ -4467,8 +4467,8 @@
 subsubsection\<open>More extension theorems\<close>
 
 lemma extension_from_clopen:
-  assumes ope: "openin (subtopology euclidean S) T"
-      and clo: "closedin (subtopology euclidean S) T"
+  assumes ope: "openin (top_of_set S) T"
+      and clo: "closedin (top_of_set S) T"
       and contf: "continuous_on T f" and fim: "f ` T \<subseteq> U" and null: "U = {} \<Longrightarrow> S = {}"
  obtains g where "continuous_on S g" "g ` S \<subseteq> U" "\<And>x. x \<in> T \<Longrightarrow> g x = f x"
 proof (cases "U = {}")
@@ -4503,8 +4503,8 @@
      and C: "C \<in> components S" and contf: "continuous_on C f" and fim: "f ` C \<subseteq> U"
  obtains g where "continuous_on S g" "g ` S \<subseteq> U" "\<And>x. x \<in> C \<Longrightarrow> g x = f x"
 proof -
-  obtain T g where ope: "openin (subtopology euclidean S) T"
-               and clo: "closedin (subtopology euclidean S) T"
+  obtain T g where ope: "openin (top_of_set S) T"
+               and clo: "closedin (top_of_set S) T"
                and "C \<subseteq> T" and contg: "continuous_on T g" and gim: "g ` T \<subseteq> U"
                and gf: "\<And>x. x \<in> C \<Longrightarrow> g x = f x"
     using S
@@ -4514,7 +4514,7 @@
       by (metis C \<open>locally connected S\<close> openin_components_locally_connected closedin_component contf fim order_refl that)
   next
     assume "compact S"
-    then obtain W g where "C \<subseteq> W" and opeW: "openin (subtopology euclidean S) W"
+    then obtain W g where "C \<subseteq> W" and opeW: "openin (top_of_set S) W"
                  and contg: "continuous_on W g"
                  and gim: "g ` W \<subseteq> U" and gf: "\<And>x. x \<in> C \<Longrightarrow> g x = f x"
       using ANR_imp_absolute_neighbourhood_extensor [of U C f S] C \<open>ANR U\<close> closedin_component contf fim by blast
@@ -4522,11 +4522,11 @@
       by (auto simp: openin_open)
     moreover have "locally compact S"
       by (simp add: \<open>compact S\<close> closed_imp_locally_compact compact_imp_closed)
-    ultimately obtain K where opeK: "openin (subtopology euclidean S) K" and "compact K" "C \<subseteq> K" "K \<subseteq> V"
+    ultimately obtain K where opeK: "openin (top_of_set S) K" and "compact K" "C \<subseteq> K" "K \<subseteq> V"
       by (metis C Int_subset_iff \<open>C \<subseteq> W\<close> \<open>compact S\<close> compact_components Sura_Bura_clopen_subset)
     show ?thesis
     proof
-      show "closedin (subtopology euclidean S) K"
+      show "closedin (top_of_set S) K"
         by (meson \<open>compact K\<close> \<open>compact S\<close> closedin_compact_eq opeK openin_imp_subset)
       show "continuous_on K g"
         by (metis Int_subset_iff V \<open>K \<subseteq> V\<close> contg continuous_on_subset opeK openin_subtopology subset_eq)
@@ -4545,13 +4545,13 @@
 lemma tube_lemma:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact S" and S: "S \<noteq> {}" "(\<lambda>x. (x,a)) ` S \<subseteq> U" 
-      and ope: "openin (subtopology euclidean (S \<times> T)) U"
-  obtains V where "openin (subtopology euclidean T) V" "a \<in> V" "S \<times> V \<subseteq> U"
+      and ope: "openin (top_of_set (S \<times> T)) U"
+  obtains V where "openin (top_of_set T) V" "a \<in> V" "S \<times> V \<subseteq> U"
 proof -
   let ?W = "{y. \<exists>x. x \<in> S \<and> (x, y) \<in> (S \<times> T - U)}"
-  have "U \<subseteq> S \<times> T" "closedin (subtopology euclidean (S \<times> T)) (S \<times> T - U)"
+  have "U \<subseteq> S \<times> T" "closedin (top_of_set (S \<times> T)) (S \<times> T - U)"
     using ope by (auto simp: openin_closedin_eq)
-  then have "closedin (subtopology euclidean T) ?W"
+  then have "closedin (top_of_set T) ?W"
     using \<open>compact S\<close> closedin_compact_projection by blast
   moreover have "a \<in> T - ?W"
     using \<open>U \<subseteq> S \<times> T\<close> S by auto
@@ -4564,16 +4564,16 @@
 lemma tube_lemma_gen:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact S" "S \<noteq> {}" "T \<subseteq> T'" "S \<times> T \<subseteq> U"
-      and ope: "openin (subtopology euclidean (S \<times> T')) U"
-  obtains V where "openin (subtopology euclidean T') V" "T \<subseteq> V" "S \<times> V \<subseteq> U"
+      and ope: "openin (top_of_set (S \<times> T')) U"
+  obtains V where "openin (top_of_set T') V" "T \<subseteq> V" "S \<times> V \<subseteq> U"
 proof -
-  have "\<And>x. x \<in> T \<Longrightarrow> \<exists>V. openin (subtopology euclidean T') V \<and> x \<in> V \<and> S \<times> V \<subseteq> U"
+  have "\<And>x. x \<in> T \<Longrightarrow> \<exists>V. openin (top_of_set T') V \<and> x \<in> V \<and> S \<times> V \<subseteq> U"
     using assms by (auto intro:  tube_lemma [OF \<open>compact S\<close>])
-  then obtain F where F: "\<And>x. x \<in> T \<Longrightarrow> openin (subtopology euclidean T') (F x) \<and> x \<in> F x \<and> S \<times> F x \<subseteq> U"
+  then obtain F where F: "\<And>x. x \<in> T \<Longrightarrow> openin (top_of_set T') (F x) \<and> x \<in> F x \<and> S \<times> F x \<subseteq> U"
     by metis
   show ?thesis
   proof
-    show "openin (subtopology euclidean T') (\<Union>(F ` T))"
+    show "openin (top_of_set T') (\<Union>(F ` T))"
       using F by blast
     show "T \<subseteq> \<Union>(F ` T)"
       using F by blast
@@ -4586,9 +4586,9 @@
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> U"
       and contg: "continuous_on S g" and gim: "g ` S \<subseteq> U"
-      and clo: "closedin (subtopology euclidean S) T"
+      and clo: "closedin (top_of_set S) T"
       and "ANR U" and hom: "homotopic_with (\<lambda>x. True) T U f g"
-    obtains V where "T \<subseteq> V" "openin (subtopology euclidean S) V"
+    obtains V where "T \<subseteq> V" "openin (top_of_set S) V"
                     "homotopic_with (\<lambda>x. True) V U f g"
 proof -
   have "T \<subseteq> S"
@@ -4604,11 +4604,11 @@
   have "continuous_on(?S0 \<union> (?S1 \<union> {0..1} \<times> T)) h'"
     unfolding h'_def
   proof (intro continuous_on_cases_local)
-    show "closedin (subtopology euclidean (?S0 \<union> (?S1 \<union> {0..1} \<times> T))) ?S0"
-         "closedin (subtopology euclidean (?S1 \<union> {0..1} \<times> T)) ?S1"
+    show "closedin (top_of_set (?S0 \<union> (?S1 \<union> {0..1} \<times> T))) ?S0"
+         "closedin (top_of_set (?S1 \<union> {0..1} \<times> T)) ?S1"
       using \<open>T \<subseteq> S\<close> by (force intro: closedin_Times closedin_subset_trans [of "{0..1} \<times> S"])+
-    show "closedin (subtopology euclidean (?S0 \<union> (?S1 \<union> {0..1} \<times> T))) (?S1 \<union> {0..1} \<times> T)"
-         "closedin (subtopology euclidean (?S1 \<union> {0..1} \<times> T)) ({0..1} \<times> T)"
+    show "closedin (top_of_set (?S0 \<union> (?S1 \<union> {0..1} \<times> T))) (?S1 \<union> {0..1} \<times> T)"
+         "closedin (top_of_set (?S1 \<union> {0..1} \<times> T)) ({0..1} \<times> T)"
       using \<open>T \<subseteq> S\<close> by (force intro: clo closedin_Times closedin_subset_trans [of "{0..1} \<times> S"])+
     show "continuous_on (?S0) (\<lambda>x. f (snd x))"
       by (intro continuous_intros continuous_on_compose2 [OF contf]) auto
@@ -4619,16 +4619,16 @@
     by (metis Sigma_Un_distrib1 Un_assoc insert_is_Un) 
   moreover have "h' ` ({0,1} \<times> S \<union> {0..1} \<times> T) \<subseteq> U"
     using fim gim him \<open>T \<subseteq> S\<close> unfolding h'_def by force
-  moreover have "closedin (subtopology euclidean ({0..1::real} \<times> S)) ({0,1} \<times> S \<union> {0..1::real} \<times> T)"
+  moreover have "closedin (top_of_set ({0..1::real} \<times> S)) ({0,1} \<times> S \<union> {0..1::real} \<times> T)"
     by (intro closedin_Times closedin_Un clo) (simp_all add: closed_subset)
   ultimately
   obtain W k where W: "({0,1} \<times> S) \<union> ({0..1} \<times> T) \<subseteq> W"
-               and opeW: "openin (subtopology euclidean ({0..1} \<times> S)) W"
+               and opeW: "openin (top_of_set ({0..1} \<times> S)) W"
                and contk: "continuous_on W k"
                and kim: "k ` W \<subseteq> U"
                and kh': "\<And>x. x \<in> ({0,1} \<times> S) \<union> ({0..1} \<times> T) \<Longrightarrow> k x = h' x"
     by (metis ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR U\<close>, of "({0,1} \<times> S) \<union> ({0..1} \<times> T)" h' "{0..1} \<times> S"])
-  obtain T' where opeT': "openin (subtopology euclidean S) T'" 
+  obtain T' where opeT': "openin (top_of_set S) T'" 
               and "T \<subseteq> T'" and TW: "{0..1} \<times> T' \<subseteq> W"
     using tube_lemma_gen [of "{0..1::real}" T S W] W \<open>T \<subseteq> S\<close> opeW by auto
   moreover have "homotopic_with (\<lambda>x. True) T' U f g"
@@ -4649,8 +4649,8 @@
 text\<open> Homotopy on a union of closed-open sets.\<close>
 proposition%unimportant homotopic_on_clopen_Union:
   fixes \<F> :: "'a::euclidean_space set set"
-  assumes "\<And>S. S \<in> \<F> \<Longrightarrow> closedin (subtopology euclidean (\<Union>\<F>)) S"
-      and "\<And>S. S \<in> \<F> \<Longrightarrow> openin (subtopology euclidean (\<Union>\<F>)) S"
+  assumes "\<And>S. S \<in> \<F> \<Longrightarrow> closedin (top_of_set (\<Union>\<F>)) S"
+      and "\<And>S. S \<in> \<F> \<Longrightarrow> openin (top_of_set (\<Union>\<F>)) S"
       and "\<And>S. S \<in> \<F> \<Longrightarrow> homotopic_with (\<lambda>x. True) S T f g"
   shows "homotopic_with (\<lambda>x. True) (\<Union>\<F>) T f g"
 proof -
@@ -4665,8 +4665,8 @@
     case False
     then obtain V :: "nat \<Rightarrow> 'a set" where V: "range V = \<V>"
       using range_from_nat_into \<open>countable \<V>\<close> by metis
-    with \<open>\<V> \<subseteq> \<F>\<close> have clo: "\<And>n. closedin (subtopology euclidean (\<Union>\<F>)) (V n)"
-                  and ope: "\<And>n. openin (subtopology euclidean (\<Union>\<F>)) (V n)"
+    with \<open>\<V> \<subseteq> \<F>\<close> have clo: "\<And>n. closedin (top_of_set (\<Union>\<F>)) (V n)"
+                  and ope: "\<And>n. openin (top_of_set (\<Union>\<F>)) (V n)"
                   and hom: "\<And>n. homotopic_with (\<lambda>x. True) (V n) T f g"
       using assms by auto 
     then obtain h where conth: "\<And>n. continuous_on ({0..1::real} \<times> V n) (h n)"
@@ -4680,20 +4680,20 @@
               and eq: "\<And>x i. \<lbrakk>x \<in> {0..1} \<times> \<Union>(V ` UNIV) \<inter>
                                    {0..1} \<times> (V i - (\<Union>m<i. V m))\<rbrakk> \<Longrightarrow> \<zeta> x = h i x"
     proof (rule pasting_lemma_exists)
-      show "{0..1} \<times> \<Union>(V ` UNIV) \<subseteq> (\<Union>i. {0..1::real} \<times> (V i - (\<Union>m<i. V m)))"
+      let ?X = "top_of_set ({0..1::real} \<times> \<Union>(range V))"
+      show "topspace ?X \<subseteq> (\<Union>i. {0..1::real} \<times> (V i - (\<Union>m<i. V m)))"
         by (force simp: Ball_def dest: wop)
-      show "openin (subtopology euclidean ({0..1} \<times> \<Union>(V ` UNIV))) 
+      show "openin (top_of_set ({0..1} \<times> \<Union>(V ` UNIV))) 
                    ({0..1::real} \<times> (V i - (\<Union>m<i. V m)))" for i
       proof (intro openin_Times openin_subtopology_self openin_diff)
-        show "openin (subtopology euclidean (\<Union>(V ` UNIV))) (V i)"
+        show "openin (top_of_set (\<Union>(V ` UNIV))) (V i)"
           using ope V eqU by auto
-        show "closedin (subtopology euclidean (\<Union>(V ` UNIV))) (\<Union>m<i. V m)"
+        show "closedin (top_of_set (\<Union>(V ` UNIV))) (\<Union>m<i. V m)"
           using V clo eqU by (force intro: closedin_Union)
       qed
-      show "continuous_on ({0..1} \<times> (V i - (\<Union>m<i. V m))) (h i)" for i
-        by (rule continuous_on_subset [OF conth]) auto
-      show "\<And>i j x. x \<in> {0..1} \<times> \<Union>(V ` UNIV) \<inter>
-                    {0..1} \<times> (V i - (\<Union>m<i. V m)) \<inter> {0..1} \<times> (V j - (\<Union>m<j. V m))
+      show "continuous_map (subtopology ?X ({0..1} \<times> (V i - \<Union> (V ` {..<i})))) euclidean (h i)"  for i
+        by (auto simp add: subtopology_subtopology intro!: continuous_on_subset [OF conth])
+      show "\<And>i j x. x \<in> topspace ?X \<inter> {0..1} \<times> (V i - (\<Union>m<i. V m)) \<inter> {0..1} \<times> (V j - (\<Union>m<j. V m))
                     \<Longrightarrow> h i x = h j x"
         by clarsimp (metis lessThan_iff linorder_neqE_nat)
     qed auto
@@ -4738,8 +4738,8 @@
       by (simp add: homotopic_with_subset_left in_components_subset)
   next
     assume R: ?rhs
-    have "\<exists>U. C \<subseteq> U \<and> closedin (subtopology euclidean S) U \<and>
-              openin (subtopology euclidean S) U \<and>
+    have "\<exists>U. C \<subseteq> U \<and> closedin (top_of_set S) U \<and>
+              openin (top_of_set S) U \<and>
               homotopic_with (\<lambda>x. True) U T f g" if C: "C \<in> components S" for C
     proof -
       have "C \<subseteq> S"
@@ -4750,9 +4750,9 @@
         assume "locally connected S"
         show ?thesis
         proof (intro exI conjI)
-          show "closedin (subtopology euclidean S) C"
+          show "closedin (top_of_set S) C"
             by (simp add: closedin_component that)
-          show "openin (subtopology euclidean S) C"
+          show "openin (top_of_set S) C"
             by (simp add: \<open>locally connected S\<close> openin_components_locally_connected that)
           show "homotopic_with (\<lambda>x. True) C T f g"
             by (simp add: R that)
@@ -4761,7 +4761,7 @@
         assume "compact S"
         have hom: "homotopic_with (\<lambda>x. True) C T f g"
           using R that by blast
-        obtain U where "C \<subseteq> U" and opeU: "openin (subtopology euclidean S) U"
+        obtain U where "C \<subseteq> U" and opeU: "openin (top_of_set S) U"
                   and hom: "homotopic_with (\<lambda>x. True) U T f g"
           using homotopic_neighbourhood_extension [OF contf fim contg gim _ \<open>ANR T\<close> hom]
             \<open>C \<in> components S\<close> closedin_component by blast
@@ -4769,11 +4769,11 @@
           by (auto simp: openin_open)
         moreover have "locally compact S"
           by (simp add: \<open>compact S\<close> closed_imp_locally_compact compact_imp_closed)
-        ultimately obtain K where opeK: "openin (subtopology euclidean S) K" and "compact K" "C \<subseteq> K" "K \<subseteq> V"
+        ultimately obtain K where opeK: "openin (top_of_set S) K" and "compact K" "C \<subseteq> K" "K \<subseteq> V"
           by (metis C Int_subset_iff Sura_Bura_clopen_subset \<open>C \<subseteq> U\<close> \<open>compact S\<close> compact_components)
         show ?thesis
         proof (intro exI conjI)
-          show "closedin (subtopology euclidean S) K"
+          show "closedin (top_of_set S) K"
             by (meson \<open>compact K\<close> \<open>compact S\<close> closedin_compact_eq opeK openin_imp_subset)
           show "homotopic_with (\<lambda>x. True) K T f g"
             using V \<open>K \<subseteq> V\<close> hom homotopic_with_subset_left opeK openin_imp_subset by fastforce
@@ -4781,8 +4781,8 @@
       qed
     qed
     then obtain \<phi> where \<phi>: "\<And>C. C \<in> components S \<Longrightarrow> C \<subseteq> \<phi> C"
-                  and clo\<phi>: "\<And>C. C \<in> components S \<Longrightarrow> closedin (subtopology euclidean S) (\<phi> C)"
-                  and ope\<phi>: "\<And>C. C \<in> components S \<Longrightarrow> openin (subtopology euclidean S) (\<phi> C)"
+                  and clo\<phi>: "\<And>C. C \<in> components S \<Longrightarrow> closedin (top_of_set S) (\<phi> C)"
+                  and ope\<phi>: "\<And>C. C \<in> components S \<Longrightarrow> openin (top_of_set S) (\<phi> C)"
                   and hom\<phi>: "\<And>C. C \<in> components S \<Longrightarrow> homotopic_with (\<lambda>x. True) (\<phi> C) T f g"
       by metis
     have Seq: "S = \<Union> (\<phi> ` components S)"
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Cauchy_Integral_Theorem.thy
--- a/src/HOL/Analysis/Cauchy_Integral_Theorem.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Cauchy_Integral_Theorem.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -7001,13 +7001,13 @@
       by (metis funpow_add o_apply)
     with that show ?thesis by auto
   qed
-  have 1: "openin (subtopology euclidean S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+  have 1: "openin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
     apply (rule open_subset, force)
     using \<open>open S\<close>
     apply (simp add: open_contains_ball Ball_def)
     apply (erule all_forward)
     using "*" by auto blast+
-  have 2: "closedin (subtopology euclidean S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+  have 2: "closedin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
     using assms
     by (auto intro: continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv)
   obtain e where "e>0" and e: "ball w e \<subseteq> S" using openE [OF \<open>open S\<close> \<open>w \<in> S\<close>] .
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Change_Of_Vars.thy
--- a/src/HOL/Analysis/Change_Of_Vars.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Change_Of_Vars.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -1567,24 +1567,24 @@
         show "countable {A. \<forall>i j. A $ i $ j \<in> \<rat>}"
           using countable_vector [OF countable_vector, of "\<lambda>i j. \<rat>"] by (simp add: countable_rat)
       qed blast
-      have *: "\<lbrakk>U \<noteq> {} \<Longrightarrow> closedin (subtopology euclidean S) (S \<inter> \<Inter> U)\<rbrakk>
-               \<Longrightarrow> closedin (subtopology euclidean S) (S \<inter> \<Inter> U)" for U
+      have *: "\<lbrakk>U \<noteq> {} \<Longrightarrow> closedin (top_of_set S) (S \<inter> \<Inter> U)\<rbrakk>
+               \<Longrightarrow> closedin (top_of_set S) (S \<inter> \<Inter> U)" for U
         by fastforce
       have eq: "{x::(real,'m)vec. P x \<and> (Q x \<longrightarrow> R x)} = {x. P x \<and> \<not> Q x} \<union> {x. P x \<and> R x}" for P Q R
         by auto
       have sets: "S \<inter> (\<Inter>y\<in>S. {x \<in> S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x)})
                   \<in> sets lebesgue" for e A d
       proof -
-        have clo: "closedin (subtopology euclidean S)
+        have clo: "closedin (top_of_set S)
                      {x \<in> S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x)}"
           for y
         proof -
           have cont1: "continuous_on S (\<lambda>x. norm (y - x))"
           and  cont2: "continuous_on S (\<lambda>x. e * norm (y - x) - norm (f y - f x - (A *v y - A *v x)))"
             by (force intro: contf continuous_intros)+
-          have clo1: "closedin (subtopology euclidean S) {x \<in> S. d \<le> norm (y - x)}"
+          have clo1: "closedin (top_of_set S) {x \<in> S. d \<le> norm (y - x)}"
             using continuous_closedin_preimage [OF cont1, of "{d..}"] by (simp add: vimage_def Int_def)
-          have clo2: "closedin (subtopology euclidean S)
+          have clo2: "closedin (top_of_set S)
                        {x \<in> S. norm (f y - f x - (A *v y - A *v x)) \<le> e * norm (y - x)}"
             using continuous_closedin_preimage [OF cont2, of "{0..}"] by (simp add: vimage_def Int_def)
           show ?thesis
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Connected.thy
--- a/src/HOL/Analysis/Connected.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Connected.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -14,8 +14,8 @@
 lemma connected_local:
  "connected S \<longleftrightarrow>
   \<not> (\<exists>e1 e2.
-      openin (subtopology euclidean S) e1 \<and>
-      openin (subtopology euclidean S) e2 \<and>
+      openin (top_of_set S) e1 \<and>
+      openin (top_of_set S) e2 \<and>
       S \<subseteq> e1 \<union> e2 \<and>
       e1 \<inter> e2 = {} \<and>
       e1 \<noteq> {} \<and>
@@ -39,8 +39,8 @@
 qed
 
 lemma connected_clopen: "connected S \<longleftrightarrow>
-  (\<forall>T. openin (subtopology euclidean S) T \<and>
-     closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
+  (\<forall>T. openin (top_of_set S) T \<and>
+     closedin (top_of_set S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
 proof -
   have "\<not> connected S \<longleftrightarrow>
     (\<exists>e1 e2. open e1 \<and> open (- e2) \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
@@ -402,7 +402,7 @@
     by blast
 qed
 
-lemma closedin_connected_component: "closedin (subtopology euclidean s) (connected_component_set s x)"
+lemma closedin_connected_component: "closedin (top_of_set s) (connected_component_set s x)"
 proof (cases "connected_component_set s x = {}")
   case True
   then show ?thesis
@@ -423,7 +423,7 @@
 qed
 
 lemma closedin_component:
-   "C \<in> components s \<Longrightarrow> closedin (subtopology euclidean s) C"
+   "C \<in> components s \<Longrightarrow> closedin (top_of_set s) C"
   using closedin_connected_component componentsE by blast
 
 
@@ -433,7 +433,7 @@
 lemma continuous_levelset_openin_cases:
   fixes f :: "_ \<Rightarrow> 'b::t1_space"
   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
-        openin (subtopology euclidean s) {x \<in> s. f x = a}
+        openin (top_of_set s) {x \<in> s. f x = a}
         \<Longrightarrow> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
   unfolding connected_clopen
   using continuous_closedin_preimage_constant by auto
@@ -441,7 +441,7 @@
 lemma continuous_levelset_openin:
   fixes f :: "_ \<Rightarrow> 'b::t1_space"
   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
-        openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
+        openin (top_of_set s) {x \<in> s. f x = a} \<Longrightarrow>
         (\<exists>x \<in> s. f x = a)  \<Longrightarrow> (\<forall>x \<in> s. f x = a)"
   using continuous_levelset_openin_cases[of s f ]
   by meson
@@ -469,8 +469,8 @@
   assumes "connected T"
       and contf: "continuous_on S f" and fim: "f ` S = T"
       and opT: "\<And>U. U \<subseteq> T
-                 \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
-                     openin (subtopology euclidean T) U"
+                 \<Longrightarrow> openin (top_of_set S) (S \<inter> f -` U) \<longleftrightarrow>
+                     openin (top_of_set T) U"
       and connT: "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f -` {y})"
     shows "connected S"
 proof (rule connectedI)
@@ -492,9 +492,9 @@
   qed
   ultimately have UU: "(S \<inter> f -` f ` (S \<inter> U)) = S \<inter> U" and VV: "(S \<inter> f -` f ` (S \<inter> V)) = S \<inter> V"
     by auto
-  have opeU: "openin (subtopology euclidean T) (f ` (S \<inter> U))"
+  have opeU: "openin (top_of_set T) (f ` (S \<inter> U))"
     by (metis UU \<open>open U\<close> fim image_Int_subset le_inf_iff opT openin_open_Int)
-  have opeV: "openin (subtopology euclidean T) (f ` (S \<inter> V))"
+  have opeV: "openin (top_of_set T) (f ` (S \<inter> V))"
     by (metis opT fim VV \<open>open V\<close> openin_open_Int image_Int_subset inf.bounded_iff)
   have "T \<subseteq> f ` (S \<inter> U) \<union> f ` (S \<inter> V)"
     using \<open>S \<subseteq> U \<union> V\<close> fim by auto
@@ -505,7 +505,7 @@
 
 lemma connected_open_monotone_preimage:
   assumes contf: "continuous_on S f" and fim: "f ` S = T"
-    and ST: "\<And>C. openin (subtopology euclidean S) C \<Longrightarrow> openin (subtopology euclidean T) (f ` C)"
+    and ST: "\<And>C. openin (top_of_set S) C \<Longrightarrow> openin (top_of_set T) (f ` C)"
     and connT: "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f -` {y})"
     and "connected C" "C \<subseteq> T"
   shows "connected (S \<inter> f -` C)"
@@ -525,12 +525,12 @@
       ultimately show ?thesis
         by metis
     qed
-    have "\<And>U. openin (subtopology euclidean (S \<inter> f -` C)) U
-               \<Longrightarrow> openin (subtopology euclidean C) (f ` U)"
+    have "\<And>U. openin (top_of_set (S \<inter> f -` C)) U
+               \<Longrightarrow> openin (top_of_set C) (f ` U)"
       using open_map_restrict [OF _ ST \<open>C \<subseteq> T\<close>] by metis
     then show "\<And>D. D \<subseteq> C
-          \<Longrightarrow> openin (subtopology euclidean (S \<inter> f -` C)) (S \<inter> f -` C \<inter> f -` D) =
-              openin (subtopology euclidean C) D"
+          \<Longrightarrow> openin (top_of_set (S \<inter> f -` C)) (S \<inter> f -` C \<inter> f -` D) =
+              openin (top_of_set C) D"
       using open_map_imp_quotient_map [of "(S \<inter> f -` C)" f] contf' by (simp add: eqC)
   qed
 qed
@@ -538,7 +538,7 @@
 
 lemma connected_closed_monotone_preimage:
   assumes contf: "continuous_on S f" and fim: "f ` S = T"
-    and ST: "\<And>C. closedin (subtopology euclidean S) C \<Longrightarrow> closedin (subtopology euclidean T) (f ` C)"
+    and ST: "\<And>C. closedin (top_of_set S) C \<Longrightarrow> closedin (top_of_set T) (f ` C)"
     and connT: "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f -` {y})"
     and "connected C" "C \<subseteq> T"
   shows "connected (S \<inter> f -` C)"
@@ -558,12 +558,12 @@
       ultimately show ?thesis
         by metis
     qed
-    have "\<And>U. closedin (subtopology euclidean (S \<inter> f -` C)) U
-               \<Longrightarrow> closedin (subtopology euclidean C) (f ` U)"
+    have "\<And>U. closedin (top_of_set (S \<inter> f -` C)) U
+               \<Longrightarrow> closedin (top_of_set C) (f ` U)"
       using closed_map_restrict [OF _ ST \<open>C \<subseteq> T\<close>] by metis
     then show "\<And>D. D \<subseteq> C
-          \<Longrightarrow> openin (subtopology euclidean (S \<inter> f -` C)) (S \<inter> f -` C \<inter> f -` D) =
-              openin (subtopology euclidean C) D"
+          \<Longrightarrow> openin (top_of_set (S \<inter> f -` C)) (S \<inter> f -` C \<inter> f -` D) =
+              openin (top_of_set C) D"
       using closed_map_imp_quotient_map [of "(S \<inter> f -` C)" f] contf' by (simp add: eqC)
   qed
 qed
@@ -576,8 +576,8 @@
 lemma connected_Un_clopen_in_complement:
   fixes S U :: "'a::metric_space set"
   assumes "connected S" "connected U" "S \<subseteq> U" 
-      and opeT: "openin (subtopology euclidean (U - S)) T" 
-      and cloT: "closedin (subtopology euclidean (U - S)) T"
+      and opeT: "openin (top_of_set (U - S)) T" 
+      and cloT: "closedin (top_of_set (U - S)) T"
     shows "connected (S \<union> T)"
 proof -
   have *: "\<lbrakk>\<And>x y. P x y \<longleftrightarrow> P y x; \<And>x y. P x y \<Longrightarrow> S \<subseteq> x \<or> S \<subseteq> y;
@@ -587,11 +587,11 @@
     unfolding connected_closedin_eq
   proof (rule *)
     fix H1 H2
-    assume H: "closedin (subtopology euclidean (S \<union> T)) H1 \<and> 
-               closedin (subtopology euclidean (S \<union> T)) H2 \<and>
+    assume H: "closedin (top_of_set (S \<union> T)) H1 \<and> 
+               closedin (top_of_set (S \<union> T)) H2 \<and>
                H1 \<union> H2 = S \<union> T \<and> H1 \<inter> H2 = {} \<and> H1 \<noteq> {} \<and> H2 \<noteq> {}"
-    then have clo: "closedin (subtopology euclidean S) (S \<inter> H1)"
-                   "closedin (subtopology euclidean S) (S \<inter> H2)"
+    then have clo: "closedin (top_of_set S) (S \<inter> H1)"
+                   "closedin (top_of_set S) (S \<inter> H2)"
       by (metis Un_upper1 closedin_closed_subset inf_commute)+
     have Seq: "S \<inter> (H1 \<union> H2) = S"
       by (simp add: H)
@@ -606,8 +606,8 @@
       using H \<open>connected S\<close> unfolding connected_closedin by blast
   next
     fix H1 H2
-    assume H: "closedin (subtopology euclidean (S \<union> T)) H1 \<and>
-               closedin (subtopology euclidean (S \<union> T)) H2 \<and>
+    assume H: "closedin (top_of_set (S \<union> T)) H1 \<and>
+               closedin (top_of_set (S \<union> T)) H2 \<and>
                H1 \<union> H2 = S \<union> T \<and> H1 \<inter> H2 = {} \<and> H1 \<noteq> {} \<and> H2 \<noteq> {}" 
        and "S \<subseteq> H1"
     then have H2T: "H2 \<subseteq> T"
@@ -616,17 +616,17 @@
       using Diff_iff opeT openin_imp_subset by auto
     with \<open>S \<subseteq> U\<close> have Ueq: "U = (U - S) \<union> (S \<union> T)" 
       by auto
-    have "openin (subtopology euclidean ((U - S) \<union> (S \<union> T))) H2"
+    have "openin (top_of_set ((U - S) \<union> (S \<union> T))) H2"
     proof (rule openin_subtopology_Un)
-      show "openin (subtopology euclidean (S \<union> T)) H2"
+      show "openin (top_of_set (S \<union> T)) H2"
         using \<open>H2 \<subseteq> T\<close> apply (auto simp: openin_closedin_eq)
         by (metis Diff_Diff_Int Diff_disjoint Diff_partition Diff_subset H Int_absorb1 Un_Diff)
-      then show "openin (subtopology euclidean (U - S)) H2"
+      then show "openin (top_of_set (U - S)) H2"
         by (meson H2T Un_upper2 opeT openin_subset_trans openin_trans)
     qed
-    moreover have "closedin (subtopology euclidean ((U - S) \<union> (S \<union> T))) H2"
+    moreover have "closedin (top_of_set ((U - S) \<union> (S \<union> T))) H2"
     proof (rule closedin_subtopology_Un)
-      show "closedin (subtopology euclidean (U - S)) H2"
+      show "closedin (top_of_set (U - S)) H2"
         using H H2T cloT closedin_subset_trans 
         by (blast intro: closedin_subtopology_Un closedin_trans)
     qed (simp add: H)
@@ -650,33 +650,33 @@
   using \<open>connected S\<close> unfolding connected_closedin_eq not_ex de_Morgan_conj
 proof clarify
   fix H3 H4 
-  assume clo3: "closedin (subtopology euclidean (U - C)) H3" 
-    and clo4: "closedin (subtopology euclidean (U - C)) H4" 
+  assume clo3: "closedin (top_of_set (U - C)) H3" 
+    and clo4: "closedin (top_of_set (U - C)) H4" 
     and "H3 \<union> H4 = U - C" and "H3 \<inter> H4 = {}" and "H3 \<noteq> {}" and "H4 \<noteq> {}"
     and * [rule_format]:
-    "\<forall>H1 H2. \<not> closedin (subtopology euclidean S) H1 \<or>
-                      \<not> closedin (subtopology euclidean S) H2 \<or>
+    "\<forall>H1 H2. \<not> closedin (top_of_set S) H1 \<or>
+                      \<not> closedin (top_of_set S) H2 \<or>
                       H1 \<union> H2 \<noteq> S \<or> H1 \<inter> H2 \<noteq> {} \<or> \<not> H1 \<noteq> {} \<or> \<not> H2 \<noteq> {}"
-  then have "H3 \<subseteq> U-C" and ope3: "openin (subtopology euclidean (U - C)) (U - C - H3)"
-    and "H4 \<subseteq> U-C" and ope4: "openin (subtopology euclidean (U - C)) (U - C - H4)"
+  then have "H3 \<subseteq> U-C" and ope3: "openin (top_of_set (U - C)) (U - C - H3)"
+    and "H4 \<subseteq> U-C" and ope4: "openin (top_of_set (U - C)) (U - C - H4)"
     by (auto simp: closedin_def)
   have "C \<noteq> {}" "C \<subseteq> U-S" "connected C"
     using C in_components_nonempty in_components_subset in_components_maximal by blast+
   have cCH3: "connected (C \<union> H3)"
   proof (rule connected_Un_clopen_in_complement [OF \<open>connected C\<close> \<open>connected U\<close> _ _ clo3])
-    show "openin (subtopology euclidean (U - C)) H3"
+    show "openin (top_of_set (U - C)) H3"
       apply (simp add: openin_closedin_eq \<open>H3 \<subseteq> U - C\<close>)
       apply (simp add: closedin_subtopology)
       by (metis Diff_cancel Diff_triv Un_Diff clo4 \<open>H3 \<inter> H4 = {}\<close> \<open>H3 \<union> H4 = U - C\<close> closedin_closed inf_commute sup_bot.left_neutral)
   qed (use clo3 \<open>C \<subseteq> U - S\<close> in auto)
   have cCH4: "connected (C \<union> H4)"
   proof (rule connected_Un_clopen_in_complement [OF \<open>connected C\<close> \<open>connected U\<close> _ _ clo4])
-    show "openin (subtopology euclidean (U - C)) H4"
+    show "openin (top_of_set (U - C)) H4"
       apply (simp add: openin_closedin_eq \<open>H4 \<subseteq> U - C\<close>)
       apply (simp add: closedin_subtopology)
       by (metis Diff_cancel Int_commute Un_Diff Un_Diff_Int \<open>H3 \<inter> H4 = {}\<close> \<open>H3 \<union> H4 = U - C\<close> clo3 closedin_closed)
   qed (use clo4 \<open>C \<subseteq> U - S\<close> in auto)
-  have "closedin (subtopology euclidean S) (S \<inter> H3)" "closedin (subtopology euclidean S) (S \<inter> H4)"
+  have "closedin (top_of_set S) (S \<inter> H3)" "closedin (top_of_set S) (S \<inter> H4)"
     using clo3 clo4 \<open>S \<subseteq> U\<close> \<open>C \<subseteq> U - S\<close> by (auto simp: closedin_closed)
   moreover have "S \<inter> H3 \<noteq> {}"      
     using components_maximal [OF C cCH3] \<open>C \<noteq> {}\<close> \<open>C \<subseteq> U - S\<close> \<open>H3 \<noteq> {}\<close> \<open>H3 \<subseteq> U - C\<close> by auto
@@ -719,8 +719,8 @@
     shows "connected S"
 proof -
   { fix t u
-    assume clt: "closedin (subtopology euclidean S) t"
-       and clu: "closedin (subtopology euclidean S) u"
+    assume clt: "closedin (top_of_set S) t"
+       and clu: "closedin (top_of_set S) u"
        and tue: "t \<inter> u = {}" and tus: "t \<union> u = S"
     have conif: "continuous_on S (\<lambda>x. if x \<in> t then 0 else 1)"
       apply (subst tus [symmetric])
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Continuous_Extension.thy
--- a/src/HOL/Analysis/Continuous_Extension.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Continuous_Extension.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -33,7 +33,7 @@
       by (simp add: supp_sum_def sum_nonneg)
     have sd_pos: "0 < setdist {x} (S - V)" if "V \<in> \<C>" "x \<in> S" "x \<in> V" for V x
     proof -
-      have "closedin (subtopology euclidean S) (S - V)"
+      have "closedin (top_of_set S) (S - V)"
         by (simp add: Diff_Diff_Int closedin_def opC openin_open_Int \<open>V \<in> \<C>\<close>)
       with that False  setdist_pos_le [of "{x}" "S - V"]
       show ?thesis
@@ -67,7 +67,7 @@
           fix x assume "x \<in> S"
           then obtain X where "open X" and x: "x \<in> S \<inter> X" and finX: "finite {U \<in> \<C>. U \<inter> X \<noteq> {}}"
             using assms by blast
-          then have OSX: "openin (subtopology euclidean S) (S \<inter> X)" by blast
+          then have OSX: "openin (top_of_set S) (S \<inter> X)" by blast
           have sumeq: "\<And>x. x \<in> S \<inter> X \<Longrightarrow>
                      (\<Sum>V | V \<in> \<C> \<and> V \<inter> X \<noteq> {}. setdist {x} (S - V))
                      = supp_sum (\<lambda>V. setdist {x} (S - V)) \<C>"
@@ -114,8 +114,8 @@
 text \<open>For Euclidean spaces the proof is easy using distances.\<close>
 
 lemma Urysohn_both_ne:
-  assumes US: "closedin (subtopology euclidean U) S"
-      and UT: "closedin (subtopology euclidean U) T"
+  assumes US: "closedin (top_of_set U) S"
+      and UT: "closedin (top_of_set U) T"
       and "S \<inter> T = {}" "S \<noteq> {}" "T \<noteq> {}" "a \<noteq> b"
   obtains f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
     where "continuous_on U f"
@@ -167,8 +167,8 @@
 qed
 
 proposition Urysohn_local_strong:
-  assumes US: "closedin (subtopology euclidean U) S"
-      and UT: "closedin (subtopology euclidean U) T"
+  assumes US: "closedin (top_of_set U) S"
+      and UT: "closedin (top_of_set U) T"
       and "S \<inter> T = {}" "a \<noteq> b"
   obtains f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
     where "continuous_on U f"
@@ -250,8 +250,8 @@
 qed
 
 lemma Urysohn_local:
-  assumes US: "closedin (subtopology euclidean U) S"
-      and UT: "closedin (subtopology euclidean U) T"
+  assumes US: "closedin (top_of_set U) S"
+      and UT: "closedin (top_of_set U) T"
       and "S \<inter> T = {}"
   obtains f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
     where "continuous_on U f"
@@ -301,7 +301,7 @@
 theorem Dugundji:
   fixes f :: "'a::{metric_space,second_countable_topology} \<Rightarrow> 'b::real_inner"
   assumes "convex C" "C \<noteq> {}"
-      and cloin: "closedin (subtopology euclidean U) S"
+      and cloin: "closedin (top_of_set U) S"
       and contf: "continuous_on S f" and "f ` S \<subseteq> C"
   obtains g where "continuous_on U g" "g ` U \<subseteq> C"
                   "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
@@ -427,7 +427,7 @@
         obtain N where N: "open N" "a \<in> N"
                    and finN: "finite {U \<in> \<C>. \<exists>a\<in>N. H U a \<noteq> 0}"
           using Hfin False \<open>a \<in> U\<close> by auto
-        have oUS: "openin (subtopology euclidean U) (U - S)"
+        have oUS: "openin (top_of_set U) (U - S)"
           using cloin by (simp add: openin_diff)
         have HcontU: "continuous (at a within U) (H T)" if "T \<in> \<C>" for T
           using Hcont [OF \<open>T \<in> \<C>\<close>] False  \<open>a \<in> U\<close> \<open>T \<in> \<C>\<close>
@@ -444,7 +444,7 @@
                     (\<lambda>x. \<Sum>T\<in>{U \<in> \<C>. \<exists>x\<in>N. H U x \<noteq> 0}. H T x *\<^sub>R f (\<A> T))"
               by (force intro: continuous_intros HcontU)+
           next
-            show "openin (subtopology euclidean U) ((U - S) \<inter> N)"
+            show "openin (top_of_set U) ((U - S) \<inter> N)"
               using N oUS openin_trans by blast
           next
             show "a \<in> (U - S) \<inter> N" using False \<open>a \<in> U\<close> N by blast
@@ -475,7 +475,7 @@
 corollary Tietze:
   fixes f :: "'a::{metric_space,second_countable_topology} \<Rightarrow> 'b::real_inner"
   assumes "continuous_on S f"
-    and "closedin (subtopology euclidean U) S"
+    and "closedin (top_of_set U) S"
     and "0 \<le> B"
     and "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> B"
   obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
@@ -485,7 +485,7 @@
 corollary%unimportant Tietze_closed_interval:
   fixes f :: "'a::{metric_space,second_countable_topology} \<Rightarrow> 'b::euclidean_space"
   assumes "continuous_on S f"
-    and "closedin (subtopology euclidean U) S"
+    and "closedin (top_of_set U) S"
     and "cbox a b \<noteq> {}"
     and "\<And>x. x \<in> S \<Longrightarrow> f x \<in> cbox a b"
   obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
@@ -496,7 +496,7 @@
 corollary%unimportant Tietze_closed_interval_1:
   fixes f :: "'a::{metric_space,second_countable_topology} \<Rightarrow> real"
   assumes "continuous_on S f"
-    and "closedin (subtopology euclidean U) S"
+    and "closedin (top_of_set U) S"
     and "a \<le> b"
     and "\<And>x. x \<in> S \<Longrightarrow> f x \<in> cbox a b"
   obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
@@ -507,7 +507,7 @@
 corollary%unimportant Tietze_open_interval:
   fixes f :: "'a::{metric_space,second_countable_topology} \<Rightarrow> 'b::euclidean_space"
   assumes "continuous_on S f"
-    and "closedin (subtopology euclidean U) S"
+    and "closedin (top_of_set U) S"
     and "box a b \<noteq> {}"
     and "\<And>x. x \<in> S \<Longrightarrow> f x \<in> box a b"
   obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
@@ -518,7 +518,7 @@
 corollary%unimportant Tietze_open_interval_1:
   fixes f :: "'a::{metric_space,second_countable_topology} \<Rightarrow> real"
   assumes "continuous_on S f"
-    and "closedin (subtopology euclidean U) S"
+    and "closedin (top_of_set U) S"
     and "a < b"
     and no: "\<And>x. x \<in> S \<Longrightarrow> f x \<in> box a b"
   obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
@@ -529,7 +529,7 @@
 corollary%unimportant Tietze_unbounded:
   fixes f :: "'a::{metric_space,second_countable_topology} \<Rightarrow> 'b::real_inner"
   assumes "continuous_on S f"
-    and "closedin (subtopology euclidean U) S"
+    and "closedin (top_of_set U) S"
   obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
   apply (rule Dugundji [of UNIV U S f])
   using assms by auto
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -62,7 +62,7 @@
     using g unfolding o_def id_def image_def by auto metis+
   show ?thesis
   proof (rule closedin_closed_trans [of "range f"])
-    show "closedin (subtopology euclidean (range f)) (f ` S)"
+    show "closedin (top_of_set (range f)) (f ` S)"
       using continuous_closedin_preimage [OF confg cgf] by simp
     show "closed (range f)"
       apply (rule closed_injective_image_subspace)
@@ -319,7 +319,7 @@
 subsection \<open>Relative interior of a set\<close>
 
 definition%important "rel_interior S =
-  {x. \<exists>T. openin (subtopology euclidean (affine hull S)) T \<and> x \<in> T \<and> T \<subseteq> S}"
+  {x. \<exists>T. openin (top_of_set (affine hull S)) T \<and> x \<in> T \<and> T \<subseteq> S}"
 
 lemma rel_interior_mono:
    "\<lbrakk>S \<subseteq> T; affine hull S = affine hull T\<rbrakk>
@@ -327,7 +327,7 @@
   by (auto simp: rel_interior_def)
 
 lemma rel_interior_maximal:
-   "\<lbrakk>T \<subseteq> S; openin(subtopology euclidean (affine hull S)) T\<rbrakk> \<Longrightarrow> T \<subseteq> (rel_interior S)"
+   "\<lbrakk>T \<subseteq> S; openin(top_of_set (affine hull S)) T\<rbrakk> \<Longrightarrow> T \<subseteq> (rel_interior S)"
   by (auto simp: rel_interior_def)
 
 lemma rel_interior:
@@ -651,20 +651,20 @@
 
 definition%important "rel_open S \<longleftrightarrow> rel_interior S = S"
 
-lemma rel_open: "rel_open S \<longleftrightarrow> openin (subtopology euclidean (affine hull S)) S"
+lemma rel_open: "rel_open S \<longleftrightarrow> openin (top_of_set (affine hull S)) S"
   unfolding rel_open_def rel_interior_def
   apply auto
-  using openin_subopen[of "subtopology euclidean (affine hull S)" S]
+  using openin_subopen[of "top_of_set (affine hull S)" S]
   apply auto
   done
 
-lemma openin_rel_interior: "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
+lemma openin_rel_interior: "openin (top_of_set (affine hull S)) (rel_interior S)"
   apply (simp add: rel_interior_def)
   apply (subst openin_subopen, blast)
   done
 
 lemma openin_set_rel_interior:
-   "openin (subtopology euclidean S) (rel_interior S)"
+   "openin (top_of_set S) (rel_interior S)"
 by (rule openin_subset_trans [OF openin_rel_interior rel_interior_subset hull_subset])
 
 lemma affine_rel_open:
@@ -806,11 +806,11 @@
 qed
 
 lemma rel_interior_eq:
-   "rel_interior s = s \<longleftrightarrow> openin(subtopology euclidean (affine hull s)) s"
+   "rel_interior s = s \<longleftrightarrow> openin(top_of_set (affine hull s)) s"
 using rel_open rel_open_def by blast
 
 lemma rel_interior_openin:
-   "openin(subtopology euclidean (affine hull s)) s \<Longrightarrow> rel_interior s = s"
+   "openin(top_of_set (affine hull s)) s \<Longrightarrow> rel_interior s = s"
 by (simp add: rel_interior_eq)
 
 lemma rel_interior_affine:
@@ -1992,9 +1992,9 @@
   proof -
     obtain e where "e > 0" and e: "cball a e \<subseteq> T"
       using \<open>open T\<close> \<open>a \<in> T\<close> by (auto simp: open_contains_cball)
-    have "openin (subtopology euclidean S) {a}"
+    have "openin (top_of_set S) {a}"
       unfolding openin_open using that \<open>a \<in> S\<close> by blast
-    moreover have "closedin (subtopology euclidean S) {a}"
+    moreover have "closedin (top_of_set S) {a}"
       by (simp add: assms)
     ultimately show "False"
       using \<open>connected S\<close> connected_clopen S by blast
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Elementary_Metric_Spaces.thy
--- a/src/HOL/Analysis/Elementary_Metric_Spaces.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Elementary_Metric_Spaces.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -2718,7 +2718,7 @@
 subsection \<open>With Abstract Topology (TODO: move and remove dependency?)\<close>
 
 lemma openin_contains_ball:
-    "openin (subtopology euclidean t) s \<longleftrightarrow>
+    "openin (top_of_set t) s \<longleftrightarrow>
      s \<subseteq> t \<and> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> ball x e \<inter> t \<subseteq> s)"
     (is "?lhs = ?rhs")
 proof
@@ -2735,7 +2735,7 @@
 qed
 
 lemma openin_contains_cball:
-   "openin (subtopology euclidean t) s \<longleftrightarrow>
+   "openin (top_of_set t) s \<longleftrightarrow>
         s \<subseteq> t \<and>
         (\<forall>x \<in> s. \<exists>e. 0 < e \<and> cball x e \<inter> t \<subseteq> s)"
   apply (simp add: openin_contains_ball)
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Elementary_Normed_Spaces.thy
--- a/src/HOL/Analysis/Elementary_Normed_Spaces.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Elementary_Normed_Spaces.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -915,7 +915,7 @@
 theorem Baire:
   fixes S::"'a::{real_normed_vector,heine_borel} set"
   assumes "closed S" "countable \<G>"
-      and ope: "\<And>T. T \<in> \<G> \<Longrightarrow> openin (subtopology euclidean S) T \<and> S \<subseteq> closure T"
+      and ope: "\<And>T. T \<in> \<G> \<Longrightarrow> openin (top_of_set S) T \<and> S \<subseteq> closure T"
  shows "S \<subseteq> closure(\<Inter>\<G>)"
 proof (cases "\<G> = {}")
   case True
@@ -930,31 +930,31 @@
   proof (clarsimp simp: closure_approachable)
     fix x and e::real
     assume "x \<in> S" "0 < e"
-    obtain TF where opeF: "\<And>n. openin (subtopology euclidean S) (TF n)"
+    obtain TF where opeF: "\<And>n. openin (top_of_set S) (TF n)"
                and ne: "\<And>n. TF n \<noteq> {}"
                and subg: "\<And>n. S \<inter> closure(TF n) \<subseteq> ?g n"
                and subball: "\<And>n. closure(TF n) \<subseteq> ball x e"
                and decr: "\<And>n. TF(Suc n) \<subseteq> TF n"
     proof -
-      have *: "\<exists>Y. (openin (subtopology euclidean S) Y \<and> Y \<noteq> {} \<and>
+      have *: "\<exists>Y. (openin (top_of_set S) Y \<and> Y \<noteq> {} \<and>
                    S \<inter> closure Y \<subseteq> ?g n \<and> closure Y \<subseteq> ball x e) \<and> Y \<subseteq> U"
-        if opeU: "openin (subtopology euclidean S) U" and "U \<noteq> {}" and cloU: "closure U \<subseteq> ball x e" for U n
+        if opeU: "openin (top_of_set S) U" and "U \<noteq> {}" and cloU: "closure U \<subseteq> ball x e" for U n
       proof -
         obtain T where T: "open T" "U = T \<inter> S"
-          using \<open>openin (subtopology euclidean S) U\<close> by (auto simp: openin_subtopology)
+          using \<open>openin (top_of_set S) U\<close> by (auto simp: openin_subtopology)
         with \<open>U \<noteq> {}\<close> have "T \<inter> closure (?g n) \<noteq> {}"
           using gin ope by fastforce
         then have "T \<inter> ?g n \<noteq> {}"
           using \<open>open T\<close> open_Int_closure_eq_empty by blast
         then obtain y where "y \<in> U" "y \<in> ?g n"
           using T ope [of "?g n", OF gin] by (blast dest:  openin_imp_subset)
-        moreover have "openin (subtopology euclidean S) (U \<inter> ?g n)"
+        moreover have "openin (top_of_set S) (U \<inter> ?g n)"
           using gin ope opeU by blast
         ultimately obtain d where U: "U \<inter> ?g n \<subseteq> S" and "d > 0" and d: "ball y d \<inter> S \<subseteq> U \<inter> ?g n"
           by (force simp: openin_contains_ball)
         show ?thesis
         proof (intro exI conjI)
-          show "openin (subtopology euclidean S) (S \<inter> ball y (d/2))"
+          show "openin (top_of_set S) (S \<inter> ball y (d/2))"
             by (simp add: openin_open_Int)
           show "S \<inter> ball y (d/2) \<noteq> {}"
             using \<open>0 < d\<close> \<open>y \<in> U\<close> opeU openin_imp_subset by fastforce
@@ -979,14 +979,14 @@
             using ball_divide_subset_numeral d by blast
         qed
       qed
-      let ?\<Phi> = "\<lambda>n X. openin (subtopology euclidean S) X \<and> X \<noteq> {} \<and>
+      let ?\<Phi> = "\<lambda>n X. openin (top_of_set S) X \<and> X \<noteq> {} \<and>
                       S \<inter> closure X \<subseteq> ?g n \<and> closure X \<subseteq> ball x e"
       have "closure (S \<inter> ball x (e / 2)) \<subseteq> closure(ball x (e/2))"
         by (simp add: closure_mono)
       also have "...  \<subseteq> ball x e"
         using \<open>e > 0\<close> by auto
       finally have "closure (S \<inter> ball x (e / 2)) \<subseteq> ball x e" .
-      moreover have"openin (subtopology euclidean S) (S \<inter> ball x (e / 2))" "S \<inter> ball x (e / 2) \<noteq> {}"
+      moreover have"openin (top_of_set S) (S \<inter> ball x (e / 2))" "S \<inter> ball x (e / 2) \<noteq> {}"
         using \<open>0 < e\<close> \<open>x \<in> S\<close> by auto
       ultimately obtain Y where Y: "?\<Phi> 0 Y \<and> Y \<subseteq> S \<inter> ball x (e / 2)"
             using * [of "S \<inter> ball x (e/2)" 0] by metis
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -2032,7 +2032,7 @@
 subsection\<open>Negligibility is a local property\<close>
 
 lemma locally_negligible_alt:
-    "negligible S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>U. openin (subtopology euclidean S) U \<and> x \<in> U \<and> negligible U)"
+    "negligible S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>U. openin (top_of_set S) U \<and> x \<in> U \<and> negligible U)"
      (is "_ = ?rhs")
 proof
   assume "negligible S"
@@ -2040,7 +2040,7 @@
     using openin_subtopology_self by blast
 next
   assume ?rhs
-  then obtain U where ope: "\<And>x. x \<in> S \<Longrightarrow> openin (subtopology euclidean S) (U x)"
+  then obtain U where ope: "\<And>x. x \<in> S \<Longrightarrow> openin (top_of_set S) (U x)"
     and cov: "\<And>x. x \<in> S \<Longrightarrow> x \<in> U x"
     and neg: "\<And>x. x \<in> S \<Longrightarrow> negligible (U x)"
     by metis
@@ -4493,7 +4493,7 @@
     fix a
     assume a: "a \<in> U n" and fa: "f a \<in> T"
     define V where "V \<equiv> {x. x \<in> U n \<and> f x \<in> T} \<inter> ball a (1 / n / 2)"
-    show "\<exists>V. openin (subtopology euclidean {x \<in> U n. f x \<in> T}) V \<and> a \<in> V \<and> negligible V"
+    show "\<exists>V. openin (top_of_set {x \<in> U n. f x \<in> T}) V \<and> a \<in> V \<and> negligible V"
     proof (intro exI conjI)
       have noxy: "norm(x - y) \<le> n * norm(f x - f y)" if "x \<in> V" "y \<in> V" for x y
         using that unfolding U_def V_def mem_Collect_eq Int_iff mem_ball dist_norm
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Function_Topology.thy
--- a/src/HOL/Analysis/Function_Topology.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Function_Topology.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -7,9 +7,9 @@
 begin
 
 
-section \<open>Product Topology\<close> (* FIX  see comments by the author *)
+section \<open>Function Topology\<close> 
 
-text \<open>We want to define the product topology.
+text \<open>We want to define the general product topology.
 
 The product topology on a product of topological spaces is generated by
 the sets which are products of open sets along finitely many coordinates, and the whole
@@ -48,7 +48,7 @@
 be the natural continuity definition of a map from the topology \<open>T1\<close> to the topology \<open>T2\<close>. Then
 the current \<open>continuous_on\<close> (with type classes) can be redefined as
 @{text [display] \<open>continuous_on s f =
-    continuous_on_topo (subtopology euclidean s) (topology euclidean) f\<close>}
+    continuous_on_topo (top_of_set s) (topology euclidean) f\<close>}
 
 In fact, I need \<open>continuous_on_topo\<close> to express the continuity of the projection on subfactors
 for the product topology, in Lemma~\<open>continuous_on_restrict_product_topology\<close>, and I show
@@ -61,369 +61,6 @@
 I realized afterwards that this theory has a lot in common with \<^file>\<open>~~/src/HOL/Library/Finite_Map.thy\<close>.
 \<close>
 
-subsection \<open>Topology without type classes\<close>
-
-subsubsection%important \<open>The topology generated by some (open) subsets\<close>
-
-text \<open>In the definition below of a generated topology, the \<open>Empty\<close> case is not necessary,
-as it follows from \<open>UN\<close> taking for \<open>K\<close> the empty set. However, it is convenient to have,
-and is never a problem in proofs, so I prefer to write it down explicitly.
-
-We do not require \<open>UNIV\<close> to be an open set, as this will not be the case in applications. (We are
-thinking of a topology on a subset of \<open>UNIV\<close>, the remaining part of \<open>UNIV\<close> being irrelevant.)\<close>
-
-inductive generate_topology_on for S where
-  Empty: "generate_topology_on S {}"
-| Int: "generate_topology_on S a \<Longrightarrow> generate_topology_on S b \<Longrightarrow> generate_topology_on S (a \<inter> b)"
-| UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology_on S k) \<Longrightarrow> generate_topology_on S (\<Union>K)"
-| Basis: "s \<in> S \<Longrightarrow> generate_topology_on S s"
-
-lemma istopology_generate_topology_on:
-  "istopology (generate_topology_on S)"
-unfolding istopology_def by (auto intro: generate_topology_on.intros)
-
-text \<open>The basic property of the topology generated by a set \<open>S\<close> is that it is the
-smallest topology containing all the elements of \<open>S\<close>:\<close>
-
-lemma generate_topology_on_coarsest:
-  assumes "istopology T"
-          "\<And>s. s \<in> S \<Longrightarrow> T s"
-          "generate_topology_on S s0"
-  shows "T s0"
-using assms(3) apply (induct rule: generate_topology_on.induct)
-using assms(1) assms(2) unfolding istopology_def by auto
-
-abbreviation%unimportant topology_generated_by::"('a set set) \<Rightarrow> ('a topology)"
-  where "topology_generated_by S \<equiv> topology (generate_topology_on S)"
-
-lemma openin_topology_generated_by_iff:
-  "openin (topology_generated_by S) s \<longleftrightarrow> generate_topology_on S s"
-  using topology_inverse'[OF istopology_generate_topology_on[of S]] by simp
-
-lemma openin_topology_generated_by:
-  "openin (topology_generated_by S) s \<Longrightarrow> generate_topology_on S s"
-using openin_topology_generated_by_iff by auto
-
-lemma topology_generated_by_topspace:
-  "topspace (topology_generated_by S) = (\<Union>S)"
-proof
-  {
-    fix s assume "openin (topology_generated_by S) s"
-    then have "generate_topology_on S s" by (rule openin_topology_generated_by)
-    then have "s \<subseteq> (\<Union>S)" by (induct, auto)
-  }
-  then show "topspace (topology_generated_by S) \<subseteq> (\<Union>S)"
-    unfolding topspace_def by auto
-next
-  have "generate_topology_on S (\<Union>S)"
-    using generate_topology_on.UN[OF generate_topology_on.Basis, of S S] by simp
-  then show "(\<Union>S) \<subseteq> topspace (topology_generated_by S)"
-    unfolding topspace_def using openin_topology_generated_by_iff by auto
-qed
-
-lemma topology_generated_by_Basis:
-  "s \<in> S \<Longrightarrow> openin (topology_generated_by S) s"
-  by (simp only: openin_topology_generated_by_iff, auto simp: generate_topology_on.Basis)
-
-lemma generate_topology_on_Inter:
-  "\<lbrakk>finite \<F>; \<And>K. K \<in> \<F> \<Longrightarrow> generate_topology_on \<S> K; \<F> \<noteq> {}\<rbrakk> \<Longrightarrow> generate_topology_on \<S> (\<Inter>\<F>)"
-  by (induction \<F> rule: finite_induct; force intro: generate_topology_on.intros)
-
-subsection\<open>Topology bases and sub-bases\<close>
-
-lemma istopology_base_alt:
-   "istopology (arbitrary union_of P) \<longleftrightarrow>
-    (\<forall>S T. (arbitrary union_of P) S \<and> (arbitrary union_of P) T
-           \<longrightarrow> (arbitrary union_of P) (S \<inter> T))"
-  by (simp add: istopology_def) (blast intro: arbitrary_union_of_Union)
-
-lemma istopology_base_eq:
-   "istopology (arbitrary union_of P) \<longleftrightarrow>
-    (\<forall>S T. P S \<and> P T \<longrightarrow> (arbitrary union_of P) (S \<inter> T))"
-  by (simp add: istopology_base_alt arbitrary_union_of_Int_eq)
-
-lemma istopology_base:
-   "(\<And>S T. \<lbrakk>P S; P T\<rbrakk> \<Longrightarrow> P(S \<inter> T)) \<Longrightarrow> istopology (arbitrary union_of P)"
-  by (simp add: arbitrary_def istopology_base_eq union_of_inc)
-
-lemma openin_topology_base_unique:
-   "openin X = arbitrary union_of P \<longleftrightarrow>
-        (\<forall>V. P V \<longrightarrow> openin X V) \<and> (\<forall>U x. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V. P V \<and> x \<in> V \<and> V \<subseteq> U))"
-   (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs
-    by (auto simp: union_of_def arbitrary_def)
-next
-  assume R: ?rhs
-  then have *: "\<exists>\<U>\<subseteq>Collect P. \<Union>\<U> = S" if "openin X S" for S
-    using that by (rule_tac x="{V. P V \<and> V \<subseteq> S}" in exI) fastforce
-  from R show ?lhs
-    by (fastforce simp add: union_of_def arbitrary_def intro: *)
-qed
-
-lemma topology_base_unique:
-   "\<lbrakk>\<And>S. P S \<Longrightarrow> openin X S;
-     \<And>U x. \<lbrakk>openin X U; x \<in> U\<rbrakk> \<Longrightarrow> \<exists>B. P B \<and> x \<in> B \<and> B \<subseteq> U\<rbrakk>
-    \<Longrightarrow> topology(arbitrary union_of P) = X"
-  by (metis openin_topology_base_unique openin_inverse [of X])
-
-lemma topology_bases_eq_aux:
-   "\<lbrakk>(arbitrary union_of P) S;
-     \<And>U x. \<lbrakk>P U; x \<in> U\<rbrakk> \<Longrightarrow> \<exists>V. Q V \<and> x \<in> V \<and> V \<subseteq> U\<rbrakk>
-        \<Longrightarrow> (arbitrary union_of Q) S"
-  by (metis arbitrary_union_of_alt arbitrary_union_of_idempot)
-
-lemma topology_bases_eq:
-   "\<lbrakk>\<And>U x. \<lbrakk>P U; x \<in> U\<rbrakk> \<Longrightarrow> \<exists>V. Q V \<and> x \<in> V \<and> V \<subseteq> U;
-    \<And>V x. \<lbrakk>Q V; x \<in> V\<rbrakk> \<Longrightarrow> \<exists>U. P U \<and> x \<in> U \<and> U \<subseteq> V\<rbrakk>
-        \<Longrightarrow> topology (arbitrary union_of P) =
-            topology (arbitrary union_of Q)"
-  by (fastforce intro:  arg_cong [where f=topology]  elim: topology_bases_eq_aux)
-
-lemma istopology_subbase:
-   "istopology (arbitrary union_of (finite intersection_of P relative_to S))"
-  by (simp add: finite_intersection_of_Int istopology_base relative_to_Int)
-
-lemma openin_subbase:
-  "openin (topology (arbitrary union_of (finite intersection_of B relative_to U))) S
-   \<longleftrightarrow> (arbitrary union_of (finite intersection_of B relative_to U)) S"
-  by (simp add: istopology_subbase topology_inverse')
-
-lemma topspace_subbase [simp]:
-   "topspace(topology (arbitrary union_of (finite intersection_of B relative_to U))) = U" (is "?lhs = _")
-proof
-  show "?lhs \<subseteq> U"
-    by (metis arbitrary_union_of_relative_to openin_subbase openin_topspace relative_to_imp_subset)
-  show "U \<subseteq> ?lhs"
-    by (metis arbitrary_union_of_inc finite_intersection_of_empty inf.orderE istopology_subbase 
-              openin_subset relative_to_inc subset_UNIV topology_inverse')
-qed
-
-lemma minimal_topology_subbase:
-   "\<lbrakk>\<And>S. P S \<Longrightarrow> openin X S; openin X U;
-     openin(topology(arbitrary union_of (finite intersection_of P relative_to U))) S\<rbrakk>
-    \<Longrightarrow> openin X S"
-  apply (simp add: istopology_subbase topology_inverse)
-  apply (simp add: union_of_def intersection_of_def relative_to_def)
-  apply (blast intro: openin_Int_Inter)
-  done
-
-lemma istopology_subbase_UNIV:
-   "istopology (arbitrary union_of (finite intersection_of P))"
-  by (simp add: istopology_base finite_intersection_of_Int)
-
-
-lemma generate_topology_on_eq:
-  "generate_topology_on S = arbitrary union_of finite' intersection_of (\<lambda>x. x \<in> S)" (is "?lhs = ?rhs")
-proof (intro ext iffI)
-  fix A
-  assume "?lhs A"
-  then show "?rhs A"
-  proof induction
-    case (Int a b)
-    then show ?case
-      by (metis (mono_tags, lifting) istopology_base_alt finite'_intersection_of_Int istopology_base)
-  next
-    case (UN K)
-    then show ?case
-      by (simp add: arbitrary_union_of_Union)
-  next
-    case (Basis s)
-    then show ?case
-      by (simp add: Sup_upper arbitrary_union_of_inc finite'_intersection_of_inc relative_to_subset)
-  qed auto
-next
-  fix A
-  assume "?rhs A"
-  then obtain \<U> where \<U>: "\<And>T. T \<in> \<U> \<Longrightarrow> \<exists>\<F>. finite' \<F> \<and> \<F> \<subseteq> S \<and> \<Inter>\<F> = T" and eq: "A = \<Union>\<U>"
-    unfolding union_of_def intersection_of_def by auto
-  show "?lhs A"
-    unfolding eq
-  proof (rule generate_topology_on.UN)
-    fix T
-    assume "T \<in> \<U>"
-    with \<U> obtain \<F> where "finite' \<F>" "\<F> \<subseteq> S" "\<Inter>\<F> = T"
-      by blast
-    have "generate_topology_on S (\<Inter>\<F>)"
-    proof (rule generate_topology_on_Inter)
-      show "finite \<F>" "\<F> \<noteq> {}"
-        by (auto simp: \<open>finite' \<F>\<close>)
-      show "\<And>K. K \<in> \<F> \<Longrightarrow> generate_topology_on S K"
-        by (metis \<open>\<F> \<subseteq> S\<close> generate_topology_on.simps subset_iff)
-    qed
-    then show "generate_topology_on S T"
-      using \<open>\<Inter>\<F> = T\<close> by blast
-  qed
-qed
-
-subsubsection%important \<open>Continuity\<close>
-
-text \<open>We will need to deal with continuous maps in terms of topologies and not in terms
-of type classes, as defined below.\<close>
-
-definition%important continuous_on_topo::"'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
-  where "continuous_on_topo T1 T2 f = ((\<forall> U. openin T2 U \<longrightarrow> openin T1 (f-`U \<inter> topspace(T1)))
-                                      \<and> (f`(topspace T1) \<subseteq> (topspace T2)))"
-
-lemma continuous_on_continuous_on_topo:
-  "continuous_on s f \<longleftrightarrow> continuous_on_topo (subtopology euclidean s) euclidean f"
-  by (auto simp: continuous_on_topo_def Int_commute continuous_openin_preimage_eq)
-
-lemma continuous_on_topo_UNIV:
-  "continuous_on UNIV f \<longleftrightarrow> continuous_on_topo euclidean euclidean f"
-using continuous_on_continuous_on_topo[of UNIV f] subtopology_UNIV[of euclidean] by auto
-
-lemma continuous_on_topo_open [intro]:
-  "continuous_on_topo T1 T2 f \<Longrightarrow> openin T2 U \<Longrightarrow> openin T1 (f-`U \<inter> topspace(T1))"
-  unfolding continuous_on_topo_def by auto
-
-lemma continuous_on_topo_topspace [intro]:
-  "continuous_on_topo T1 T2 f \<Longrightarrow> f`(topspace T1) \<subseteq> (topspace T2)"
-unfolding continuous_on_topo_def by auto
-
-lemma continuous_on_generated_topo_iff:
-  "continuous_on_topo T1 (topology_generated_by S) f \<longleftrightarrow>
-      ((\<forall>U. U \<in> S \<longrightarrow> openin T1 (f-`U \<inter> topspace(T1))) \<and> (f`(topspace T1) \<subseteq> (\<Union> S)))"
-unfolding continuous_on_topo_def topology_generated_by_topspace
-proof (auto simp add: topology_generated_by_Basis)
-  assume H: "\<forall>U. U \<in> S \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)"
-  fix U assume "openin (topology_generated_by S) U"
-  then have "generate_topology_on S U" by (rule openin_topology_generated_by)
-  then show "openin T1 (f -` U \<inter> topspace T1)"
-  proof (induct)
-    fix a b
-    assume H: "openin T1 (f -` a \<inter> topspace T1)" "openin T1 (f -` b \<inter> topspace T1)"
-    have "f -` (a \<inter> b) \<inter> topspace T1 = (f-`a \<inter> topspace T1) \<inter> (f-`b \<inter> topspace T1)"
-      by auto
-    then show "openin T1 (f -` (a \<inter> b) \<inter> topspace T1)" using H by auto
-  next
-    fix K
-    assume H: "openin T1 (f -` k \<inter> topspace T1)" if "k\<in> K" for k
-    define L where "L = {f -` k \<inter> topspace T1|k. k \<in> K}"
-    have *: "openin T1 l" if "l \<in>L" for l using that H unfolding L_def by auto
-    have "openin T1 (\<Union>L)" using openin_Union[OF *] by simp
-    moreover have "(\<Union>L) = (f -` \<Union>K \<inter> topspace T1)" unfolding L_def by auto
-    ultimately show "openin T1 (f -` \<Union>K \<inter> topspace T1)" by simp
-  qed (auto simp add: H)
-qed
-
-lemma continuous_on_generated_topo:
-  assumes "\<And>U. U \<in>S \<Longrightarrow> openin T1 (f-`U \<inter> topspace(T1))"
-          "f`(topspace T1) \<subseteq> (\<Union> S)"
-  shows "continuous_on_topo T1 (topology_generated_by S) f"
-  using assms continuous_on_generated_topo_iff by blast
-
-proposition continuous_on_topo_compose:
-  assumes "continuous_on_topo T1 T2 f" "continuous_on_topo T2 T3 g"
-  shows "continuous_on_topo T1 T3 (g o f)"
-  using assms unfolding continuous_on_topo_def
-proof (auto)
-  fix U :: "'c set"
-  assume H: "openin T3 U"
-  have "openin T1 (f -` (g -` U \<inter> topspace T2) \<inter> topspace T1)"
-    using H assms by blast
-  moreover have "f -` (g -` U \<inter> topspace T2) \<inter> topspace T1 = (g \<circ> f) -` U \<inter> topspace T1"
-    using H assms continuous_on_topo_topspace by fastforce
-  ultimately show "openin T1 ((g \<circ> f) -` U \<inter> topspace T1)"
-    by simp
-qed (blast)
-
-lemma continuous_on_topo_preimage_topspace [intro]:
-  assumes "continuous_on_topo T1 T2 f"
-  shows "f-`(topspace T2) \<inter> topspace T1 = topspace T1"
-using assms unfolding continuous_on_topo_def by auto
-
-
-subsubsection%important \<open>Pullback topology\<close>
-
-text \<open>Pulling back a topology by map gives again a topology. \<open>subtopology\<close> is
-a special case of this notion, pulling back by the identity. We introduce the general notion as
-we will need it to define the strong operator topology on the space of continuous linear operators,
-by pulling back the product topology on the space of all functions.\<close>
-
-text \<open>\<open>pullback_topology A f T\<close> is the pullback of the topology \<open>T\<close> by the map \<open>f\<close> on
-the set \<open>A\<close>.\<close>
-
-definition%important pullback_topology::"('a set) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b topology) \<Rightarrow> ('a topology)"
-  where "pullback_topology A f T = topology (\<lambda>S. \<exists>U. openin T U \<and> S = f-`U \<inter> A)"
-
-lemma istopology_pullback_topology:
-  "istopology (\<lambda>S. \<exists>U. openin T U \<and> S = f-`U \<inter> A)"
-  unfolding istopology_def proof (auto)
-  fix K assume "\<forall>S\<in>K. \<exists>U. openin T U \<and> S = f -` U \<inter> A"
-  then have "\<exists>U. \<forall>S\<in>K. openin T (U S) \<and> S = f-`(U S) \<inter> A"
-    by (rule bchoice)
-  then obtain U where U: "\<forall>S\<in>K. openin T (U S) \<and> S = f-`(U S) \<inter> A"
-    by blast
-  define V where "V = (\<Union>S\<in>K. U S)"
-  have "openin T V" "\<Union>K = f -` V \<inter> A" unfolding V_def using U by auto
-  then show "\<exists>V. openin T V \<and> \<Union>K = f -` V \<inter> A" by auto
-qed
-
-lemma openin_pullback_topology:
-  "openin (pullback_topology A f T) S \<longleftrightarrow> (\<exists>U. openin T U \<and> S = f-`U \<inter> A)"
-unfolding pullback_topology_def topology_inverse'[OF istopology_pullback_topology] by auto
-
-lemma topspace_pullback_topology:
-  "topspace (pullback_topology A f T) = f-`(topspace T) \<inter> A"
-by (auto simp add: topspace_def openin_pullback_topology)
-
-proposition continuous_on_topo_pullback [intro]:
-  assumes "continuous_on_topo T1 T2 g"
-  shows "continuous_on_topo (pullback_topology A f T1) T2 (g o f)"
-unfolding continuous_on_topo_def
-proof (auto)
-  fix U::"'b set" assume "openin T2 U"
-  then have "openin T1 (g-`U \<inter> topspace T1)"
-    using assms unfolding continuous_on_topo_def by auto
-  have "(g o f)-`U \<inter> topspace (pullback_topology A f T1) = (g o f)-`U \<inter> A \<inter> f-`(topspace T1)"
-    unfolding topspace_pullback_topology by auto
-  also have "... = f-`(g-`U \<inter> topspace T1) \<inter> A "
-    by auto
-  also have "openin (pullback_topology A f T1) (...)"
-    unfolding openin_pullback_topology using \<open>openin T1 (g-`U \<inter> topspace T1)\<close> by auto
-  finally show "openin (pullback_topology A f T1) ((g \<circ> f) -` U \<inter> topspace (pullback_topology A f T1))"
-    by auto
-next
-  fix x assume "x \<in> topspace (pullback_topology A f T1)"
-  then have "f x \<in> topspace T1"
-    unfolding topspace_pullback_topology by auto
-  then show "g (f x) \<in> topspace T2"
-    using assms unfolding continuous_on_topo_def by auto
-qed
-
-proposition continuous_on_topo_pullback' [intro]:
-  assumes "continuous_on_topo T1 T2 (f o g)" "topspace T1 \<subseteq> g-`A"
-  shows "continuous_on_topo T1 (pullback_topology A f T2) g"
-unfolding continuous_on_topo_def
-proof (auto)
-  fix U assume "openin (pullback_topology A f T2) U"
-  then have "\<exists>V. openin T2 V \<and> U = f-`V \<inter> A"
-    unfolding openin_pullback_topology by auto
-  then obtain V where "openin T2 V" "U = f-`V \<inter> A"
-    by blast
-  then have "g -` U \<inter> topspace T1 = g-`(f-`V \<inter> A) \<inter> topspace T1"
-    by blast
-  also have "... = (f o g)-`V \<inter> (g-`A \<inter> topspace T1)"
-    by auto
-  also have "... = (f o g)-`V \<inter> topspace T1"
-    using assms(2) by auto
-  also have "openin T1 (...)"
-    using assms(1) \<open>openin T2 V\<close> by auto
-  finally show "openin T1 (g -` U \<inter> topspace T1)" by simp
-next
-  fix x assume "x \<in> topspace T1"
-  have "(f o g) x \<in> topspace T2"
-    using assms(1) \<open>x \<in> topspace T1\<close> unfolding continuous_on_topo_def by auto
-  then have "g x \<in> f-`(topspace T2)"
-    unfolding comp_def by blast
-  moreover have "g x \<in> A" using assms(2) \<open>x \<in> topspace T1\<close> by blast
-  ultimately show "g x \<in> topspace (pullback_topology A f T2)"
-    unfolding topspace_pullback_topology by blast
-qed
-
-
 subsection \<open>The product topology\<close>
 
 text \<open>We can now define the product topology, as generated by
@@ -605,6 +242,9 @@
   then show ?thesis using \<open>x \<in> U\<close> by auto
 qed
 
+lemma product_topology_empty_discrete:
+   "product_topology T {} = discrete_topology {(\<lambda>x. undefined)}"
+  by (simp add: subtopology_eq_discrete_topology_sing)
 
 lemma openin_product_topology:
    "openin (product_topology X I) =
@@ -793,6 +433,15 @@
   using PiE_singleton closedin_product_topology [of X I]
   by (metis (no_types, lifting) all_not_in_conv insertI1)
 
+lemma product_topology_empty:
+   "product_topology X {} = topology (\<lambda>S. S \<in> {{},{\<lambda>k. undefined}})"
+  unfolding product_topology union_of_def intersection_of_def arbitrary_def relative_to_def
+  by (auto intro: arg_cong [where f=topology])
+
+lemma openin_product_topology_empty: "openin (product_topology X {}) S \<longleftrightarrow> S \<in> {{},{\<lambda>k. undefined}}"
+  unfolding union_of_def intersection_of_def arbitrary_def relative_to_def openin_product_topology
+  by auto
+
 
 subsubsection \<open>The basic property of the product topology is the continuity of projections:\<close>
 
@@ -949,10 +598,10 @@
   by (rule continuous_on_topo_product_coordinates, simp)
 
 lemma continuous_on_coordinatewise_then_product [intro, continuous_intros]:
-  fixes f :: "('a \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"
+  fixes f :: "'a::topological_space \<Rightarrow> 'b \<Rightarrow> 'c::topological_space"
   assumes "\<And>i. continuous_on S (\<lambda>x. f x i)"
   shows "continuous_on S f"
-  using continuous_on_topo_coordinatewise_then_product [of UNIV, where T = "\<lambda>i. euclideanreal"]
+  using continuous_on_topo_coordinatewise_then_product [of UNIV, where T = "\<lambda>i. euclidean"]
   by (metis UNIV_I assms continuous_on_continuous_on_topo euclidean_product_topology)
 
 lemma continuous_on_product_then_coordinatewise_UNIV:
@@ -1583,4 +1232,354 @@
 instance "fun" :: (countable, polish_space) polish_space
 by standard
 
+
+subsection\<open>The Alexander subbase theorem\<close>
+
+theorem Alexander_subbase:
+  assumes X: "topology (arbitrary union_of (finite intersection_of (\<lambda>x. x \<in> \<B>) relative_to \<Union>\<B>)) = X"
+    and fin: "\<And>C. \<lbrakk>C \<subseteq> \<B>; \<Union>C = topspace X\<rbrakk> \<Longrightarrow> \<exists>C'. finite C' \<and> C' \<subseteq> C \<and> \<Union>C' = topspace X"
+  shows "compact_space X"
+proof -
+  have U\<B>: "\<Union>\<B> = topspace X"
+    by (simp flip: X)
+  have False if \<U>: "\<forall>U\<in>\<U>. openin X U" and sub: "topspace X \<subseteq> \<Union>\<U>"
+    and neg: "\<And>\<F>. \<lbrakk>\<F> \<subseteq> \<U>; finite \<F>\<rbrakk> \<Longrightarrow> \<not> topspace X \<subseteq> \<Union>\<F>" for \<U>
+  proof -
+    define \<A> where "\<A> \<equiv> {\<C>. (\<forall>U \<in> \<C>. openin X U) \<and> topspace X \<subseteq> \<Union>\<C> \<and> (\<forall>\<F>. finite \<F> \<longrightarrow> \<F> \<subseteq> \<C> \<longrightarrow> ~(topspace X \<subseteq> \<Union>\<F>))}"
+    have 1: "\<A> \<noteq> {}"
+      unfolding \<A>_def using sub \<U> neg by force
+    have 2: "\<Union>\<C> \<in> \<A>" if "\<C>\<noteq>{}" and \<C>: "subset.chain \<A> \<C>" for \<C>
+      unfolding \<A>_def
+    proof (intro CollectI conjI ballI allI impI notI)
+      show "openin X U" if U: "U \<in> \<Union>\<C>" for U
+        using U \<C> unfolding \<A>_def subset_chain_def by force
+      have "\<C> \<subseteq> \<A>"
+        using subset_chain_def \<C> by blast
+      with that \<A>_def show UUC: "topspace X \<subseteq> \<Union>(\<Union>\<C>)"
+        by blast
+      show "False" if "finite \<F>" and "\<F> \<subseteq> \<Union>\<C>" and "topspace X \<subseteq> \<Union>\<F>" for \<F>
+      proof -
+        obtain \<B> where "\<B> \<in> \<C>" "\<F> \<subseteq> \<B>"
+          by (metis Sup_empty \<C> \<open>\<F> \<subseteq> \<Union>\<C>\<close> \<open>finite \<F>\<close> UUC empty_subsetI finite.emptyI finite_subset_Union_chain neg)
+        then show False
+          using \<A>_def \<open>\<C> \<subseteq> \<A>\<close> \<open>finite \<F>\<close> \<open>topspace X \<subseteq> \<Union>\<F>\<close> by blast
+      qed
+    qed
+    obtain \<K> where "\<K> \<in> \<A>" and "\<And>X. \<lbrakk>X\<in>\<A>; \<K> \<subseteq> X\<rbrakk> \<Longrightarrow> X = \<K>"
+      using subset_Zorn_nonempty [OF 1 2] by metis
+    then have *: "\<And>\<W>. \<lbrakk>\<And>W. W\<in>\<W> \<Longrightarrow> openin X W; topspace X \<subseteq> \<Union>\<W>; \<K> \<subseteq> \<W>;
+                        \<And>\<F>. \<lbrakk>finite \<F>; \<F> \<subseteq> \<W>; topspace X \<subseteq> \<Union>\<F>\<rbrakk> \<Longrightarrow> False\<rbrakk>
+           \<Longrightarrow> \<W> = \<K>"
+      and ope: "\<forall>U\<in>\<K>. openin X U" and top: "topspace X \<subseteq> \<Union>\<K>"
+      and non: "\<And>\<F>. \<lbrakk>finite \<F>; \<F> \<subseteq> \<K>; topspace X \<subseteq> \<Union>\<F>\<rbrakk> \<Longrightarrow> False"
+      unfolding \<A>_def by simp_all metis+
+    then obtain x where "x \<in> topspace X" "x \<notin> \<Union>(\<B> \<inter> \<K>)"
+    proof -
+      have "\<Union>(\<B> \<inter> \<K>) \<noteq> \<Union>\<B>"
+        by (metis \<open>\<Union>\<B> = topspace X\<close> fin inf.bounded_iff non order_refl)
+      then have "\<exists>a. a \<notin> \<Union>(\<B> \<inter> \<K>) \<and> a \<in> \<Union>\<B>"
+        by blast
+      then show ?thesis
+        using that by (metis U\<B>)
+    qed
+    obtain C where C: "openin X C" "C \<in> \<K>" "x \<in> C"
+      using \<open>x \<in> topspace X\<close> ope top by auto
+    then have "C \<subseteq> topspace X"
+      by (metis openin_subset)
+    then have "(arbitrary union_of (finite intersection_of (\<lambda>x. x \<in> \<B>) relative_to \<Union>\<B>)) C"
+      using openin_subbase C unfolding X [symmetric] by blast
+    moreover have "C \<noteq> topspace X"
+      using \<open>\<K> \<in> \<A>\<close> \<open>C \<in> \<K>\<close> unfolding \<A>_def by blast
+    ultimately obtain \<V> W where W: "(finite intersection_of (\<lambda>x. x \<in> \<B>) relative_to topspace X) W"
+      and "x \<in> W" "W \<in> \<V>" "\<Union>\<V> \<noteq> topspace X" "C = \<Union>\<V>"
+      using C  by (auto simp: union_of_def U\<B>)
+    then have "\<Union>\<V> \<subseteq> topspace X"
+      by (metis \<open>C \<subseteq> topspace X\<close>)
+    then have "topspace X \<notin> \<V>"
+      using \<open>\<Union>\<V> \<noteq> topspace X\<close> by blast
+    then obtain \<B>' where \<B>': "finite \<B>'" "\<B>' \<subseteq> \<B>" "x \<in> \<Inter>\<B>'" "W = topspace X \<inter> \<Inter>\<B>'"
+      using W \<open>x \<in> W\<close> unfolding relative_to_def intersection_of_def by auto
+    then have "\<Inter>\<B>' \<subseteq> \<Union>\<B>"
+      using \<open>W \<in> \<V>\<close> \<open>\<Union>\<V> \<noteq> topspace X\<close> \<open>\<Union>\<V> \<subseteq> topspace X\<close> by blast
+    then have "\<Inter>\<B>' \<subseteq> C"
+      using U\<B> \<open>C = \<Union>\<V>\<close> \<open>W = topspace X \<inter> \<Inter>\<B>'\<close> \<open>W \<in> \<V>\<close> by auto
+    have "\<forall>b \<in> \<B>'. \<exists>C'. finite C' \<and> C' \<subseteq> \<K> \<and> topspace X \<subseteq> \<Union>(insert b C')"
+    proof
+      fix b
+      assume "b \<in> \<B>'"
+      have "insert b \<K> = \<K>" if neg: "\<not> (\<exists>C'. finite C' \<and> C' \<subseteq> \<K> \<and> topspace X \<subseteq> \<Union>(insert b C'))"
+      proof (rule *)
+        show "openin X W" if "W \<in> insert b \<K>" for W
+          using that
+        proof
+          have "b \<in> \<B>"
+            using \<open>b \<in> \<B>'\<close> \<open>\<B>' \<subseteq> \<B>\<close> by blast
+          then have "\<exists>\<U>. finite \<U> \<and> \<U> \<subseteq> \<B> \<and> \<Inter>\<U> = b"
+            by (rule_tac x="{b}" in exI) auto
+          moreover have "\<Union>\<B> \<inter> b = b"
+            using \<B>'(2) \<open>b \<in> \<B>'\<close> by auto
+          ultimately show "openin X W" if "W = b"
+            using  that \<open>b \<in> \<B>'\<close>
+            apply (simp add: openin_subbase flip: X)
+            apply (auto simp: arbitrary_def intersection_of_def relative_to_def intro!: union_of_inc)
+            done
+          show "openin X W" if "W \<in> \<K>"
+            by (simp add: \<open>W \<in> \<K>\<close> ope)
+        qed
+      next
+        show "topspace X \<subseteq> \<Union> (insert b \<K>)"
+          using top by auto
+      next
+        show False if "finite \<F>" and "\<F> \<subseteq> insert b \<K>" "topspace X \<subseteq> \<Union>\<F>" for \<F>
+        proof -
+          have "insert b (\<F> \<inter> \<K>) = \<F>"
+            using non that by blast
+          then show False
+            by (metis Int_lower2 finite_insert neg that(1) that(3))
+        qed
+      qed auto
+      then show "\<exists>C'. finite C' \<and> C' \<subseteq> \<K> \<and> topspace X \<subseteq> \<Union>(insert b C')"
+        using \<open>b \<in> \<B>'\<close> \<open>x \<notin> \<Union>(\<B> \<inter> \<K>)\<close> \<B>'
+        by (metis IntI InterE Union_iff subsetD insertI1)
+    qed
+    then obtain F where F: "\<forall>b \<in> \<B>'. finite (F b) \<and> F b \<subseteq> \<K> \<and> topspace X \<subseteq> \<Union>(insert b (F b))"
+      by metis
+    let ?\<D> = "insert C (\<Union>(F ` \<B>'))"
+    show False
+    proof (rule non)
+      have "topspace X \<subseteq> (\<Inter>b \<in> \<B>'. \<Union>(insert b (F b)))"
+        using F by (simp add: INT_greatest)
+      also have "\<dots> \<subseteq> \<Union>?\<D>"
+        using \<open>\<Inter>\<B>' \<subseteq> C\<close> by force
+      finally show "topspace X \<subseteq> \<Union>?\<D>" .
+      show "?\<D> \<subseteq> \<K>"
+        using \<open>C \<in> \<K>\<close> F by auto
+      show "finite ?\<D>"
+        using \<open>finite \<B>'\<close> F by auto
+    qed
+  qed
+  then show ?thesis
+    by (force simp: compact_space_def compactin_def)
+qed
+
+
+corollary Alexander_subbase_alt:
+  assumes "U \<subseteq> \<Union>\<B>"
+  and fin: "\<And>C. \<lbrakk>C \<subseteq> \<B>; U \<subseteq> \<Union>C\<rbrakk> \<Longrightarrow> \<exists>C'. finite C' \<and> C' \<subseteq> C \<and> U \<subseteq> \<Union>C'"
+   and X: "topology
+              (arbitrary union_of
+                   (finite intersection_of (\<lambda>x. x \<in> \<B>) relative_to U)) = X"
+ shows "compact_space X"
+proof -
+  have "topspace X = U"
+    using X topspace_subbase by fastforce
+  have eq: "\<Union> (Collect ((\<lambda>x. x \<in> \<B>) relative_to U)) = U"
+    unfolding relative_to_def
+    using \<open>U \<subseteq> \<Union>\<B>\<close> by blast
+  have *: "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<C> \<and> \<Union>\<F> = topspace X"
+    if  "\<C> \<subseteq> Collect ((\<lambda>x. x \<in> \<B>) relative_to topspace X)" and UC: "\<Union>\<C> = topspace X" for \<C>
+  proof -
+    have "\<C> \<subseteq> (\<lambda>U. topspace X \<inter> U) ` \<B>"
+      using that by (auto simp: relative_to_def)
+    then obtain \<B>' where "\<B>' \<subseteq> \<B>" and \<B>': "\<C> = (\<inter>) (topspace X) ` \<B>'"
+      by (auto simp: subset_image_iff)
+    moreover have "U \<subseteq> \<Union>\<B>'"
+      using \<B>' \<open>topspace X = U\<close> UC by auto
+    ultimately obtain \<C>' where "finite \<C>'" "\<C>' \<subseteq> \<B>'" "U \<subseteq> \<Union>\<C>'"
+      using fin [of \<B>'] \<open>topspace X = U\<close> \<open>U \<subseteq> \<Union>\<B>\<close> by blast
+    then show ?thesis
+      unfolding \<B>' exists_finite_subset_image \<open>topspace X = U\<close> by auto
+  qed
+  show ?thesis
+    apply (rule Alexander_subbase [where \<B> = "Collect ((\<lambda>x. x \<in> \<B>) relative_to (topspace X))"])
+     apply (simp flip: X)
+     apply (metis finite_intersection_of_relative_to eq)
+    apply (blast intro: *)
+    done
+qed
+
+proposition continuous_map_componentwise:
+   "continuous_map X (product_topology Y I) f \<longleftrightarrow>
+    f ` (topspace X) \<subseteq> extensional I \<and> (\<forall>k \<in> I. continuous_map X (Y k) (\<lambda>x. f x k))"
+    (is "?lhs \<longleftrightarrow> _ \<and> ?rhs")
+proof (cases "\<forall>x \<in> topspace X. f x \<in> extensional I")
+  case True
+  then have "f ` (topspace X) \<subseteq> extensional I"
+    by force
+  moreover have ?rhs if L: ?lhs
+  proof -
+    have "openin X {x \<in> topspace X. f x k \<in> U}" if "k \<in> I" and "openin (Y k) U" for k U
+    proof -
+      have "openin (product_topology Y I) ({Y. Y k \<in> U} \<inter> (\<Pi>\<^sub>E i\<in>I. topspace (Y i)))"
+        apply (simp add: openin_product_topology flip: arbitrary_union_of_relative_to)
+        apply (simp add: relative_to_def)
+        using that apply (blast intro: arbitrary_union_of_inc finite_intersection_of_inc)
+        done
+      with that have "openin X {x \<in> topspace X. f x \<in> ({Y. Y k \<in> U} \<inter> (\<Pi>\<^sub>E i\<in>I. topspace (Y i)))}"
+        using L unfolding continuous_map_def by blast
+      moreover have "{x \<in> topspace X. f x \<in> ({Y. Y k \<in> U} \<inter> (\<Pi>\<^sub>E i\<in>I. topspace (Y i)))} = {x \<in> topspace X. f x k \<in> U}"
+        using L by (auto simp: continuous_map_def)
+      ultimately show ?thesis
+        by metis
+    qed
+    with that
+    show ?thesis
+      by (auto simp: continuous_map_def)
+  qed
+  moreover have ?lhs if ?rhs
+  proof -
+    have 1: "\<And>x. x \<in> topspace X \<Longrightarrow> f x \<in> (\<Pi>\<^sub>E i\<in>I. topspace (Y i))"
+      using that True by (auto simp: continuous_map_def PiE_iff)
+    have 2: "{x \<in> S. \<exists>T\<in>\<T>. f x \<in> T} = (\<Union>T\<in>\<T>. {x \<in> S. f x \<in> T})" for S \<T>
+      by blast
+    have 3: "{x \<in> S. \<forall>U\<in>\<U>. f x \<in> U} = (\<Inter> (insert S ((\<lambda>U. {x \<in> S. f x \<in> U}) ` \<U>)))" for S \<U>
+      by blast
+    show ?thesis
+      unfolding continuous_map_def openin_product_topology arbitrary_def
+    proof (clarsimp simp: all_union_of 1 2)
+      fix \<T>
+      assume \<T>: "\<T> \<subseteq> Collect (finite intersection_of (\<lambda>F. \<exists>i U. F = {f. f i \<in> U} \<and> i \<in> I \<and> openin (Y i) U)
+                  relative_to (\<Pi>\<^sub>E i\<in>I. topspace (Y i)))"
+      show "openin X (\<Union>T\<in>\<T>. {x \<in> topspace X. f x \<in> T})"
+      proof (rule openin_Union; clarify)
+        fix S T
+        assume "T \<in> \<T>"
+        obtain \<U> where "T = (\<Pi>\<^sub>E i\<in>I. topspace (Y i)) \<inter> \<Inter>\<U>" and "finite \<U>"
+          "\<U> \<subseteq> {{f. f i \<in> U} |i U. i \<in> I \<and> openin (Y i) U}"
+          using subsetD [OF \<T> \<open>T \<in> \<T>\<close>] by (auto simp: intersection_of_def relative_to_def)
+        with that show "openin X {x \<in> topspace X. f x \<in> T}"
+          apply (simp add: continuous_map_def 1 cong: conj_cong)
+          unfolding 3
+          apply (rule openin_Inter; auto)
+          done
+      qed
+    qed
+  qed
+  ultimately show ?thesis
+    by metis
+next
+  case False
+  then show ?thesis
+    by (auto simp: continuous_map_def PiE_def)
+qed
+
+
+lemma continuous_map_componentwise_UNIV:
+   "continuous_map X (product_topology Y UNIV) f \<longleftrightarrow> (\<forall>k. continuous_map X (Y k) (\<lambda>x. f x k))"
+  by (simp add: continuous_map_componentwise)
+
+lemma continuous_map_product_projection [continuous_intros]:
+   "k \<in> I \<Longrightarrow> continuous_map (product_topology X I) (X k) (\<lambda>x. x k)"
+  using continuous_map_componentwise [of "product_topology X I" X I id] by simp
+
+proposition open_map_product_projection:
+  assumes "i \<in> I"
+  shows "open_map (product_topology Y I) (Y i) (\<lambda>f. f i)"
+  unfolding openin_product_topology all_union_of arbitrary_def open_map_def image_Union
+proof clarify
+  fix \<V>
+  assume \<V>: "\<V> \<subseteq> Collect
+               (finite intersection_of
+                (\<lambda>F. \<exists>i U. F = {f. f i \<in> U} \<and> i \<in> I \<and> openin (Y i) U) relative_to
+                           topspace (product_topology Y I))"
+  show "openin (Y i) (\<Union>x\<in>\<V>. (\<lambda>f. f i) ` x)"
+  proof (rule openin_Union, clarify)
+    fix S V
+    assume "V \<in> \<V>"
+    obtain \<F> where "finite \<F>"
+      and V: "V = (\<Pi>\<^sub>E i\<in>I. topspace (Y i)) \<inter> \<Inter>\<F>"
+      and \<F>: "\<F> \<subseteq> {{f. f i \<in> U} |i U. i \<in> I \<and> openin (Y i) U}"
+      using subsetD [OF \<V> \<open>V \<in> \<V>\<close>]
+      by (auto simp: intersection_of_def relative_to_def)
+    show "openin (Y i) ((\<lambda>f. f i) ` V)"
+    proof (subst openin_subopen; clarify)
+      fix x f
+      assume "f \<in> V"
+      let ?T = "{a \<in> topspace(Y i).
+                   (\<lambda>j. if j = i then a
+                        else if j \<in> I then f j else undefined) \<in> (\<Pi>\<^sub>E i\<in>I. topspace (Y i)) \<inter> \<Inter>\<F>}"
+      show "\<exists>T. openin (Y i) T \<and> f i \<in> T \<and> T \<subseteq> (\<lambda>f. f i) ` V"
+      proof (intro exI conjI)
+        show "openin (Y i) ?T"
+        proof (rule openin_continuous_map_preimage)
+          have "continuous_map (Y i) (Y k) (\<lambda>x. if k = i then x else f k)" if "k \<in> I" for k
+          proof (cases "k=i")
+            case True
+            then show ?thesis
+              by (metis (mono_tags) continuous_map_id eq_id_iff)
+          next
+            case False
+            then show ?thesis
+              by simp (metis IntD1 PiE_iff V \<open>f \<in> V\<close> that)
+          qed
+          then show "continuous_map (Y i) (product_topology Y I)
+                  (\<lambda>x j. if j = i then x else if j \<in> I then f j else undefined)"
+            by (auto simp: continuous_map_componentwise assms extensional_def)
+        next
+          have "openin (product_topology Y I) (\<Pi>\<^sub>E i\<in>I. topspace (Y i))"
+            by (metis openin_topspace topspace_product_topology)
+          moreover have "openin (product_topology Y I) (\<Inter>B\<in>\<F>. (\<Pi>\<^sub>E i\<in>I. topspace (Y i)) \<inter> B)"
+                         if "\<F> \<noteq> {}"
+          proof -
+            show ?thesis
+            proof (rule openin_Inter)
+              show "\<And>X. X \<in> (\<inter>) (\<Pi>\<^sub>E i\<in>I. topspace (Y i)) ` \<F> \<Longrightarrow> openin (product_topology Y I) X"
+                unfolding openin_product_topology relative_to_def
+                apply (clarify intro!: arbitrary_union_of_inc)
+                apply (rename_tac F)
+                apply (rule_tac x=F in exI)
+                using subsetD [OF \<F>]
+                apply (force intro: finite_intersection_of_inc)
+                done
+            qed (use \<open>finite \<F>\<close> \<open>\<F> \<noteq> {}\<close> in auto)
+          qed
+          ultimately show "openin (product_topology Y I) ((\<Pi>\<^sub>E i\<in>I. topspace (Y i)) \<inter> \<Inter>\<F>)"
+            by (auto simp only: Int_Inter_eq split: if_split)
+        qed
+      next
+        have eqf: "(\<lambda>j. if j = i then f i else if j \<in> I then f j else undefined) = f"
+          using PiE_arb V \<open>f \<in> V\<close> by force
+        show "f i \<in> ?T"
+          using V assms \<open>f \<in> V\<close> by (auto simp: PiE_iff eqf)
+      next
+        show "?T \<subseteq> (\<lambda>f. f i) ` V"
+          unfolding V by (auto simp: intro!: rev_image_eqI)
+      qed
+    qed
+  qed
+qed
+
+lemma retraction_map_product_projection:
+  assumes "i \<in> I"
+  shows "(retraction_map (product_topology X I) (X i) (\<lambda>x. x i) \<longleftrightarrow>
+         (topspace (product_topology X I) = {}) \<longrightarrow> topspace (X i) = {})"
+  (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+    using retraction_imp_surjective_map by force
+next
+  assume R: ?rhs
+  show ?lhs
+  proof (cases "topspace (product_topology X I) = {}")
+    case True
+    then show ?thesis
+      using R by (auto simp: retraction_map_def retraction_maps_def continuous_map_on_empty)
+  next
+    case False
+    have *: "\<exists>g. continuous_map (X i) (product_topology X I) g \<and> (\<forall>x\<in>topspace (X i). g x i = x)"
+      if z: "z \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i))" for z
+    proof -
+      have cm: "continuous_map (X i) (X j) (\<lambda>x. if j = i then x else z j)" if "j \<in> I" for j
+        using \<open>j \<in> I\<close> z  by (case_tac "j = i") auto
+      show ?thesis
+        using \<open>i \<in> I\<close> that
+        by (rule_tac x="\<lambda>x j. if j = i then x else z j" in exI) (auto simp: continuous_map_componentwise PiE_iff extensional_def cm)
+    qed
+    show ?thesis
+      using \<open>i \<in> I\<close> False
+      by (auto simp: retraction_map_def retraction_maps_def assms continuous_map_product_projection *)
+  qed
+qed
+
 end
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Further_Topology.thy
--- a/src/HOL/Analysis/Further_Topology.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Further_Topology.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -1149,7 +1149,7 @@
   assumes "finite K" "affine U" and contf: "continuous_on (U - K) f"
       and fim: "f ` (U - K) \<subseteq> T"
       and comps: "\<And>C. \<lbrakk>C \<in> components(U - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
-      and clo: "closedin (subtopology euclidean U) S" and K: "disjnt K S" "K \<subseteq> U"
+      and clo: "closedin (top_of_set U) S" and K: "disjnt K S" "K \<subseteq> U"
   obtains g where "continuous_on (U - L) g" "g ` (U - L) \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
 proof (cases "K = {}")
   case True
@@ -1176,11 +1176,11 @@
     have "C \<subseteq> U-S" "C \<inter> L \<noteq> {}"
       by (simp_all add: in_components_subset comps that)
     then obtain a where a: "a \<in> C" "a \<in> L" by auto
-    have opeUC: "openin (subtopology euclidean U) C"
+    have opeUC: "openin (top_of_set U) C"
     proof (rule openin_trans)
-      show "openin (subtopology euclidean (U-S)) C"
+      show "openin (top_of_set (U-S)) C"
         by (simp add: \<open>locally connected U\<close> clo locally_diff_closed openin_components_locally_connected [OF _ C])
-      show "openin (subtopology euclidean U) (U - S)"
+      show "openin (top_of_set U) (U - S)"
         by (simp add: clo openin_diff)
     qed
     then obtain d where "C \<subseteq> U" "0 < d" and d: "cball a d \<inter> U \<subseteq> C"
@@ -1192,10 +1192,10 @@
                  and bou: "bounded {x. (\<not> (h x = x \<and> k x = x))}"
                  and hin: "\<And>x. x \<in> C \<inter> K \<Longrightarrow> h x \<in> ball a d \<inter> U"
     proof (rule homeomorphism_grouping_points_exists_gen [of C "ball a d \<inter> U" "C \<inter> K" "S \<union> C"])
-      show "openin (subtopology euclidean C) (ball a d \<inter> U)"
+      show "openin (top_of_set C) (ball a d \<inter> U)"
         by (metis open_ball \<open>C \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> inf.absorb_iff2 inf.orderE inf_assoc open_openin openin_subtopology)
-      show "openin (subtopology euclidean (affine hull C)) C"
-        by (metis \<open>a \<in> C\<close> \<open>openin (subtopology euclidean U) C\<close> affine_hull_eq affine_hull_openin all_not_in_conv \<open>affine U\<close>)
+      show "openin (top_of_set (affine hull C)) C"
+        by (metis \<open>a \<in> C\<close> \<open>openin (top_of_set U) C\<close> affine_hull_eq affine_hull_openin all_not_in_conv \<open>affine U\<close>)
       show "ball a d \<inter> U \<noteq> {}"
         using \<open>0 < d\<close> \<open>C \<subseteq> U\<close> \<open>a \<in> C\<close> by force
       show "finite (C \<inter> K)"
@@ -1323,13 +1323,14 @@
   obtain g where contg: "continuous_on (S \<union> UF) g"
              and gh: "\<And>x i. \<lbrakk>i \<in> F; x \<in> (S \<union> UF) \<inter> (S \<union> (i - {a i}))\<rbrakk>
                             \<Longrightarrow> g x = h i x"
-  proof (rule pasting_lemma_exists_closed [OF \<open>finite F\<close>, of "S \<union> UF" "\<lambda>C. S \<union> (C - {a C})" h])
-    show "S \<union> UF \<subseteq> (\<Union>C\<in>F. S \<union> (C - {a C}))"
+  proof (rule pasting_lemma_exists_closed [OF \<open>finite F\<close>])
+    let ?X = "top_of_set (S \<union> UF)"
+    show "topspace ?X \<subseteq> (\<Union>C\<in>F. S \<union> (C - {a C}))"
       using \<open>C0 \<in> F\<close> by (force simp: UF_def)
-    show "closedin (subtopology euclidean (S \<union> UF)) (S \<union> (C - {a C}))"
+    show "closedin (top_of_set (S \<union> UF)) (S \<union> (C - {a C}))"
          if "C \<in> F" for C
     proof (rule closedin_closed_subset [of U "S \<union> C"])
-      show "closedin (subtopology euclidean U) (S \<union> C)"
+      show "closedin (top_of_set U) (S \<union> C)"
         apply (rule closedin_Un_complement_component [OF \<open>locally connected U\<close> clo])
         using F_def that by blast
     next
@@ -1346,22 +1347,26 @@
       show "S \<union> (C - {a C}) = (S \<union> C) \<inter> (S \<union> UF)"
         using F_def UF_def components_nonoverlap that by auto
     qed
-  next
-    show "continuous_on (S \<union> (C' - {a C'})) (h C')" if "C' \<in> F" for C'
-      using ah F_def that by blast
+    show "continuous_map (subtopology ?X (S \<union> (C' - {a C'}))) euclidean (h C')" if "C' \<in> F" for C'
+    proof -
+      have C': "C' \<in> components (U - S)" "C' \<inter> K \<noteq> {}"
+        using F_def that by blast+
+      show ?thesis
+        using ah [OF C'] by (auto simp: F_def subtopology_subtopology intro: continuous_on_subset)
+    qed
     show "\<And>i j x. \<lbrakk>i \<in> F; j \<in> F;
-                   x \<in> (S \<union> UF) \<inter> (S \<union> (i - {a i})) \<inter> (S \<union> (j - {a j}))\<rbrakk>
+                   x \<in> topspace ?X \<inter> (S \<union> (i - {a i})) \<inter> (S \<union> (j - {a j}))\<rbrakk>
                   \<Longrightarrow> h i x = h j x"
       using components_eq by (fastforce simp: components_eq F_def ah)
-  qed blast
+  qed auto
   have SU': "S \<union> \<Union>G \<union> (S \<union> UF) \<subseteq> U"
     using \<open>S \<subseteq> U\<close> in_components_subset by (auto simp: F_def G_def UF_def)
-  have clo1: "closedin (subtopology euclidean (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> \<Union>G)"
+  have clo1: "closedin (top_of_set (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> \<Union>G)"
   proof (rule closedin_closed_subset [OF _ SU'])
-    have *: "\<And>C. C \<in> F \<Longrightarrow> openin (subtopology euclidean U) C"
+    have *: "\<And>C. C \<in> F \<Longrightarrow> openin (top_of_set U) C"
       unfolding F_def
       by clarify (metis (no_types, lifting) \<open>locally connected U\<close> clo closedin_def locally_diff_closed openin_components_locally_connected openin_trans topspace_euclidean_subtopology)
-    show "closedin (subtopology euclidean U) (U - UF)"
+    show "closedin (top_of_set U) (U - UF)"
       unfolding UF_def
       by (force intro: openin_delete *)
     show "S \<union> \<Union>G = (U - UF) \<inter> (S \<union> \<Union>G \<union> (S \<union> UF))"
@@ -1370,9 +1375,9 @@
        apply (metis DiffD1 UnionI Union_components)
       by (metis (no_types, lifting) IntI components_nonoverlap empty_iff)
   qed
-  have clo2: "closedin (subtopology euclidean (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> UF)"
+  have clo2: "closedin (top_of_set (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> UF)"
   proof (rule closedin_closed_subset [OF _ SU'])
-    show "closedin (subtopology euclidean U) (\<Union>C\<in>F. S \<union> C)"
+    show "closedin (top_of_set U) (\<Union>C\<in>F. S \<union> C)"
       apply (rule closedin_Union)
        apply (simp add: \<open>finite F\<close>)
       using F_def \<open>locally connected U\<close> clo closedin_Un_complement_component by blast
@@ -1467,7 +1472,7 @@
              and him: "h ` (T - \<xi> ` K) \<subseteq> rel_frontier U"
              and hg: "\<And>x. x \<in> S \<Longrightarrow> h x = g x"
   proof (rule extend_map_affine_to_sphere1 [OF \<open>finite K\<close> \<open>affine T\<close> contg gim, of S "\<xi> ` K"])
-    show cloTS: "closedin (subtopology euclidean T) S"
+    show cloTS: "closedin (top_of_set T) S"
       by (simp add: \<open>compact S\<close> \<open>S \<subseteq> T\<close> closed_subset compact_imp_closed)
     show "\<And>C. \<lbrakk>C \<in> components (T - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> \<xi> ` K \<noteq> {}"
       using \<xi> components_eq by blast
@@ -2008,8 +2013,8 @@
   fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
   assumes contf: "continuous_on U f" and injf: "inj_on f U" and fim: "f ` U \<subseteq> S"
       and "subspace S" and dimS: "dim S = DIM('b::euclidean_space)"
-      and ope: "openin (subtopology euclidean S) U"
-    shows "openin (subtopology euclidean S) (f ` U)"
+      and ope: "openin (top_of_set S) U"
+    shows "openin (top_of_set S) (f ` U)"
 proof -
   have "U \<subseteq> S"
     using ope openin_imp_subset by blast
@@ -2033,7 +2038,7 @@
         apply (rule continuous_on_subset [OF homeomorphism_cont2 [OF homhk]])
         using fim homhk homeomorphism_apply2 ope openin_subset by fastforce
     qed
-    have ope_iff: "\<And>T. open T \<longleftrightarrow> openin (subtopology euclidean (k ` S)) T"
+    have ope_iff: "\<And>T. open T \<longleftrightarrow> openin (top_of_set (k ` S)) T"
       using homhk homeomorphism_image2 open_openin by fastforce
     show "open (k ` U)"
       by (simp add: ope_iff homeomorphism_imp_open_map [OF homkh ope])
@@ -2053,14 +2058,14 @@
   fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
   assumes contf: "continuous_on U f" and injf: "inj_on f U" and fim: "f ` U \<subseteq> S"
       and "subspace S"
-      and ope: "openin (subtopology euclidean S) U"
-    shows "openin (subtopology euclidean S) (f ` U)"
+      and ope: "openin (top_of_set S) U"
+    shows "openin (top_of_set S) (f ` U)"
 proof -
   define S' where "S' \<equiv> {y. \<forall>x \<in> S. orthogonal x y}"
   have "subspace S'"
     by (simp add: S'_def subspace_orthogonal_to_vectors)
   define g where "g \<equiv> \<lambda>z::'a*'a. ((f \<circ> fst)z, snd z)"
-  have "openin (subtopology euclidean (S \<times> S')) (g ` (U \<times> S'))"
+  have "openin (top_of_set (S \<times> S')) (g ` (U \<times> S'))"
   proof (rule inv_of_domain_ss0)
     show "continuous_on (U \<times> S') g"
       apply (simp add: g_def)
@@ -2072,7 +2077,7 @@
       using injf by (auto simp: g_def inj_on_def)
     show "subspace (S \<times> S')"
       by (simp add: \<open>subspace S'\<close> \<open>subspace S\<close> subspace_Times)
-    show "openin (subtopology euclidean (S \<times> S')) (U \<times> S')"
+    show "openin (top_of_set (S \<times> S')) (U \<times> S')"
       by (simp add: openin_Times [OF ope])
     have "dim (S \<times> S') = dim S + dim S'"
       by (simp add: \<open>subspace S'\<close> \<open>subspace S\<close> dim_Times)
@@ -2092,11 +2097,11 @@
 
 corollary invariance_of_domain_subspaces:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes ope: "openin (subtopology euclidean U) S"
+  assumes ope: "openin (top_of_set U) S"
       and "subspace U" "subspace V" and VU: "dim V \<le> dim U"
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
       and injf: "inj_on f S"
-    shows "openin (subtopology euclidean V) (f ` S)"
+    shows "openin (top_of_set V) (f ` S)"
 proof -
   obtain V' where "subspace V'" "V' \<subseteq> U" "dim V' = dim V"
     using choose_subspace_of_subspace [OF VU]
@@ -2119,7 +2124,7 @@
       by (simp add: homeomorphism_symD homhk)
     have hfV': "(h \<circ> f) ` S \<subseteq> V'"
       using fim homeomorphism_image1 homhk by fastforce
-    moreover have "openin (subtopology euclidean U) ((h \<circ> f) ` S)"
+    moreover have "openin (top_of_set U) ((h \<circ> f) ` S)"
     proof (rule inv_of_domain_ss1)
       show "continuous_on S (h \<circ> f)"
         by (meson contf continuous_on_compose continuous_on_subset fim homeomorphism_cont1 homhk)
@@ -2129,14 +2134,14 @@
       show "(h \<circ> f) ` S \<subseteq> U"
         using \<open>V' \<subseteq> U\<close> hfV' by auto
       qed (auto simp: assms)
-    ultimately show "openin (subtopology euclidean V') ((h \<circ> f) ` S)"
+    ultimately show "openin (top_of_set V') ((h \<circ> f) ` S)"
       using openin_subset_trans \<open>V' \<subseteq> U\<close> by force
   qed
 qed
 
 corollary invariance_of_dimension_subspaces:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes ope: "openin (subtopology euclidean U) S"
+  assumes ope: "openin (top_of_set U) S"
       and "subspace U" "subspace V"
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
       and injf: "inj_on f S" and "S \<noteq> {}"
@@ -2158,7 +2163,7 @@
     moreover have "inj_on (h \<circ> f) S"
       apply (clarsimp simp: inj_on_def)
       by (metis fim homeomorphism_apply1 homhk image_subset_iff inj_onD injf)
-    ultimately have ope_hf: "openin (subtopology euclidean U) ((h \<circ> f) ` S)"
+    ultimately have ope_hf: "openin (top_of_set U) ((h \<circ> f) ` S)"
       using invariance_of_domain_subspaces [OF ope \<open>subspace U\<close> \<open>subspace U\<close>] by blast
     have "(h \<circ> f) ` S \<subseteq> T"
       using fim homeomorphism_image1 homhk by fastforce
@@ -2176,11 +2181,11 @@
 
 corollary invariance_of_domain_affine_sets:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes ope: "openin (subtopology euclidean U) S"
+  assumes ope: "openin (top_of_set U) S"
       and aff: "affine U" "affine V" "aff_dim V \<le> aff_dim U"
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
       and injf: "inj_on f S"
-    shows "openin (subtopology euclidean V) (f ` S)"
+    shows "openin (top_of_set V) (f ` S)"
 proof (cases "S = {}")
   case True
   then show ?thesis by auto
@@ -2188,9 +2193,9 @@
   case False
   obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
     using False fim ope openin_contains_cball by fastforce
-  have "openin (subtopology euclidean ((+) (- b) ` V)) (((+) (- b) \<circ> f \<circ> (+) a) ` (+) (- a) ` S)"
+  have "openin (top_of_set ((+) (- b) ` V)) (((+) (- b) \<circ> f \<circ> (+) a) ` (+) (- a) ` S)"
   proof (rule invariance_of_domain_subspaces)
-    show "openin (subtopology euclidean ((+) (- a) ` U)) ((+) (- a) ` S)"
+    show "openin (top_of_set ((+) (- a) ` U)) ((+) (- a) ` S)"
       by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
     show "subspace ((+) (- a) ` U)"
       by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace_subtract \<open>affine U\<close> cong: image_cong_simp)
@@ -2211,7 +2216,7 @@
 
 corollary invariance_of_dimension_affine_sets:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes ope: "openin (subtopology euclidean U) S"
+  assumes ope: "openin (top_of_set U) S"
       and aff: "affine U" "affine V"
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
       and injf: "inj_on f S" and "S \<noteq> {}"
@@ -2221,7 +2226,7 @@
     using \<open>S \<noteq> {}\<close> fim ope openin_contains_cball by fastforce
   have "dim ((+) (- a) ` U) \<le> dim ((+) (- b) ` V)"
   proof (rule invariance_of_dimension_subspaces)
-    show "openin (subtopology euclidean ((+) (- a) ` U)) ((+) (- a) ` S)"
+    show "openin (top_of_set ((+) (- a) ` U)) ((+) (- a) ` S)"
       by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
     show "subspace ((+) (- a) ` U)"
       by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace_subtract \<open>affine U\<close> cong: image_cong_simp)
@@ -2269,7 +2274,7 @@
   case False
   have "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
   proof (rule invariance_of_dimension_affine_sets)
-    show "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
+    show "openin (top_of_set (affine hull S)) (rel_interior S)"
       by (simp add: openin_rel_interior)
     show "continuous_on (rel_interior S) f"
       using contf continuous_on_subset rel_interior_subset by blast
@@ -2335,7 +2340,7 @@
   assume "open T"
   have eq: "f ` S \<inter> g -` T = f ` (S \<inter> T)"
     by (auto simp: gf)
-  show "openin (subtopology euclidean (f ` S)) (f ` S \<inter> g -` T)"
+  show "openin (top_of_set (f ` S)) (f ` S \<inter> g -` T)"
     apply (subst eq)
     apply (rule open_openin_trans)
       apply (rule invariance_of_domain_gen)
@@ -2530,9 +2535,9 @@
 proof (rule rel_interior_maximal)
   show "f ` rel_interior S \<subseteq> f ` S"
     by(simp add: image_mono rel_interior_subset)
-  show "openin (subtopology euclidean (affine hull f ` S)) (f ` rel_interior S)"
+  show "openin (top_of_set (affine hull f ` S)) (f ` rel_interior S)"
   proof (rule invariance_of_domain_affine_sets)
-    show "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
+    show "openin (top_of_set (affine hull S)) (rel_interior S)"
       by (simp add: openin_rel_interior)
     show "aff_dim (affine hull f ` S) \<le> aff_dim (affine hull S)"
       by (metis aff_dim_affine_hull aff_dim_subset fim TS order_trans)
@@ -2826,8 +2831,8 @@
   assumes contf: "continuous_on S f" and injf: "inj_on f S" and fim: "f ` S \<subseteq> T"
       and U: "bounded U" "convex U"
       and "affine T" and affTU: "aff_dim T < aff_dim U"
-      and ope: "openin (subtopology euclidean (rel_frontier U)) S"
-   shows "openin (subtopology euclidean T) (f ` S)"
+      and ope: "openin (top_of_set (rel_frontier U)) S"
+   shows "openin (top_of_set T) (f ` S)"
 proof (cases "rel_frontier U = {}")
   case True
   then show ?thesis
@@ -2857,9 +2862,9 @@
     by (simp_all add: homeomorphism_def subset_eq)
   have [simp]: "aff_dim T \<le> aff_dim V"
     by (simp add: affTU affV)
-  have "openin (subtopology euclidean T) ((f \<circ> h) ` g ` (S - {b}))"
+  have "openin (top_of_set T) ((f \<circ> h) ` g ` (S - {b}))"
   proof (rule invariance_of_domain_affine_sets [OF _ \<open>affine V\<close>])
-    show "openin (subtopology euclidean V) (g ` (S - {b}))"
+    show "openin (top_of_set V) (g ` (S - {b}))"
       apply (rule homeomorphism_imp_open_map [OF gh])
       by (meson Diff_mono Diff_subset SU ope openin_delete openin_subset_trans order_refl)
     show "continuous_on (g ` (S - {b})) (f \<circ> h)"
@@ -2874,9 +2879,9 @@
       by (metis fim image_comp image_mono hgsub subset_trans)
   qed (auto simp: assms)
   moreover
-  have "openin (subtopology euclidean T) ((f \<circ> k) ` j ` (S - {c}))"
+  have "openin (top_of_set T) ((f \<circ> k) ` j ` (S - {c}))"
   proof (rule invariance_of_domain_affine_sets [OF _ \<open>affine V\<close>])
-    show "openin (subtopology euclidean V) (j ` (S - {c}))"
+    show "openin (top_of_set V) (j ` (S - {c}))"
       apply (rule homeomorphism_imp_open_map [OF jk])
       by (meson Diff_mono Diff_subset SU ope openin_delete openin_subset_trans order_refl)
     show "continuous_on (j ` (S - {c})) (f \<circ> k)"
@@ -2890,7 +2895,7 @@
     show "(f \<circ> k) ` j ` (S - {c}) \<subseteq> T"
       by (metis fim image_comp image_mono kjsub subset_trans)
   qed (auto simp: assms)
-  ultimately have "openin (subtopology euclidean T) ((f \<circ> h) ` g ` (S - {b}) \<union> ((f \<circ> k) ` j ` (S - {c})))"
+  ultimately have "openin (top_of_set T) ((f \<circ> h) ` g ` (S - {b}) \<union> ((f \<circ> k) ` j ` (S - {c})))"
     by (rule openin_Un)
   moreover have "(f \<circ> h) ` g ` (S - {b}) = f ` (S - {b})"
   proof -
@@ -2931,8 +2936,8 @@
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on S f" and injf: "inj_on f S" and fim: "f ` S \<subseteq> T"
       and "r \<noteq> 0" "affine T" and affTU: "aff_dim T < DIM('a)"
-      and ope: "openin (subtopology euclidean (sphere a r)) S"
-   shows "openin (subtopology euclidean T) (f ` S)"
+      and ope: "openin (top_of_set (sphere a r)) S"
+   shows "openin (top_of_set T) (f ` S)"
 proof (cases "sphere a r = {}")
   case True
   then show ?thesis
@@ -2943,7 +2948,7 @@
   proof (rule invariance_of_domain_sphere_affine_set_gen [OF contf injf fim bounded_cball convex_cball \<open>affine T\<close>])
     show "aff_dim T < aff_dim (cball a r)"
       by (metis False affTU aff_dim_cball assms(4) linorder_cases sphere_empty)
-    show "openin (subtopology euclidean (rel_frontier (cball a r))) S"
+    show "openin (top_of_set (rel_frontier (cball a r))) S"
       by (simp add: \<open>r \<noteq> 0\<close> ope)
   qed
 qed
@@ -3379,7 +3384,7 @@
     by (rule continuous_on_exp [OF continuous_on_id])
   show "range exp = - {0::complex}"
     by auto (metis exp_Ln range_eqI)
-  show "\<exists>T. z \<in> T \<and> openin (subtopology euclidean (- {0})) T \<and>
+  show "\<exists>T. z \<in> T \<and> openin (top_of_set (- {0})) T \<and>
              (\<exists>v. \<Union>v = exp -` T \<and> (\<forall>u\<in>v. open u) \<and> disjoint v \<and>
                   (\<forall>u\<in>v. \<exists>q. homeomorphism u T exp q))"
         if "z \<in> - {0::complex}" for z
@@ -3399,7 +3404,7 @@
       moreover have "inj_on exp (ball (Ln z) 1)"
         apply (rule inj_on_subset [OF inj_on_exp_pi [of "Ln z"]])
         using pi_ge_two by (simp add: ball_subset_ball_iff)
-      ultimately show "openin (subtopology euclidean (- {0})) (exp ` ball (Ln z) 1)"
+      ultimately show "openin (top_of_set (- {0})) (exp ` ball (Ln z) 1)"
         by (auto simp: openin_open_eq invariance_of_domain continuous_on_exp [OF continuous_on_id])
       show "\<Union>\<V> = exp -` exp ` ball (Ln z) 1"
         by (force simp: \<V>_def Complex_Transcendental.exp_eq image_iff)
@@ -3647,150 +3652,150 @@
 text\<open>Many similar proofs below.\<close>
 lemma upper_hemicontinuous:
   assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<subseteq> T"
-    shows "((\<forall>U. openin (subtopology euclidean T) U
-                 \<longrightarrow> openin (subtopology euclidean S) {x \<in> S. f x \<subseteq> U}) \<longleftrightarrow>
-            (\<forall>U. closedin (subtopology euclidean T) U
-                 \<longrightarrow> closedin (subtopology euclidean S) {x \<in> S. f x \<inter> U \<noteq> {}}))"
+    shows "((\<forall>U. openin (top_of_set T) U
+                 \<longrightarrow> openin (top_of_set S) {x \<in> S. f x \<subseteq> U}) \<longleftrightarrow>
+            (\<forall>U. closedin (top_of_set T) U
+                 \<longrightarrow> closedin (top_of_set S) {x \<in> S. f x \<inter> U \<noteq> {}}))"
           (is "?lhs = ?rhs")
 proof (intro iffI allI impI)
   fix U
-  assume * [rule_format]: ?lhs and "closedin (subtopology euclidean T) U"
-  then have "openin (subtopology euclidean T) (T - U)"
+  assume * [rule_format]: ?lhs and "closedin (top_of_set T) U"
+  then have "openin (top_of_set T) (T - U)"
     by (simp add: openin_diff)
-  then have "openin (subtopology euclidean S) {x \<in> S. f x \<subseteq> T - U}"
+  then have "openin (top_of_set S) {x \<in> S. f x \<subseteq> T - U}"
     using * [of "T-U"] by blast
   moreover have "S - {x \<in> S. f x \<subseteq> T - U} = {x \<in> S. f x \<inter> U \<noteq> {}}"
     using assms by blast
-  ultimately show "closedin (subtopology euclidean S) {x \<in> S. f x \<inter> U \<noteq> {}}"
+  ultimately show "closedin (top_of_set S) {x \<in> S. f x \<inter> U \<noteq> {}}"
     by (simp add: openin_closedin_eq)
 next
   fix U
-  assume * [rule_format]: ?rhs and "openin (subtopology euclidean T) U"
-  then have "closedin (subtopology euclidean T) (T - U)"
+  assume * [rule_format]: ?rhs and "openin (top_of_set T) U"
+  then have "closedin (top_of_set T) (T - U)"
     by (simp add: closedin_diff)
-  then have "closedin (subtopology euclidean S) {x \<in> S. f x \<inter> (T - U) \<noteq> {}}"
+  then have "closedin (top_of_set S) {x \<in> S. f x \<inter> (T - U) \<noteq> {}}"
     using * [of "T-U"] by blast
   moreover have "{x \<in> S. f x \<inter> (T - U) \<noteq> {}} = S - {x \<in> S. f x \<subseteq> U}"
     using assms by auto
-  ultimately show "openin (subtopology euclidean S) {x \<in> S. f x \<subseteq> U}"
+  ultimately show "openin (top_of_set S) {x \<in> S. f x \<subseteq> U}"
     by (simp add: openin_closedin_eq)
 qed
 
 lemma lower_hemicontinuous:
   assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<subseteq> T"
-    shows "((\<forall>U. closedin (subtopology euclidean T) U
-                 \<longrightarrow> closedin (subtopology euclidean S) {x \<in> S. f x \<subseteq> U}) \<longleftrightarrow>
-            (\<forall>U. openin (subtopology euclidean T) U
-                 \<longrightarrow> openin (subtopology euclidean S) {x \<in> S. f x \<inter> U \<noteq> {}}))"
+    shows "((\<forall>U. closedin (top_of_set T) U
+                 \<longrightarrow> closedin (top_of_set S) {x \<in> S. f x \<subseteq> U}) \<longleftrightarrow>
+            (\<forall>U. openin (top_of_set T) U
+                 \<longrightarrow> openin (top_of_set S) {x \<in> S. f x \<inter> U \<noteq> {}}))"
           (is "?lhs = ?rhs")
 proof (intro iffI allI impI)
   fix U
-  assume * [rule_format]: ?lhs and "openin (subtopology euclidean T) U"
-  then have "closedin (subtopology euclidean T) (T - U)"
+  assume * [rule_format]: ?lhs and "openin (top_of_set T) U"
+  then have "closedin (top_of_set T) (T - U)"
     by (simp add: closedin_diff)
-  then have "closedin (subtopology euclidean S) {x \<in> S. f x \<subseteq> T-U}"
+  then have "closedin (top_of_set S) {x \<in> S. f x \<subseteq> T-U}"
     using * [of "T-U"] by blast
   moreover have "{x \<in> S. f x \<subseteq> T-U} = S - {x \<in> S. f x \<inter> U \<noteq> {}}"
     using assms by auto
-  ultimately show "openin (subtopology euclidean S) {x \<in> S. f x \<inter> U \<noteq> {}}"
+  ultimately show "openin (top_of_set S) {x \<in> S. f x \<inter> U \<noteq> {}}"
     by (simp add: openin_closedin_eq)
 next
   fix U
-  assume * [rule_format]: ?rhs and "closedin (subtopology euclidean T) U"
-  then have "openin (subtopology euclidean T) (T - U)"
+  assume * [rule_format]: ?rhs and "closedin (top_of_set T) U"
+  then have "openin (top_of_set T) (T - U)"
     by (simp add: openin_diff)
-  then have "openin (subtopology euclidean S) {x \<in> S. f x \<inter> (T - U) \<noteq> {}}"
+  then have "openin (top_of_set S) {x \<in> S. f x \<inter> (T - U) \<noteq> {}}"
     using * [of "T-U"] by blast
   moreover have "S - {x \<in> S. f x \<inter> (T - U) \<noteq> {}} = {x \<in> S. f x \<subseteq> U}"
     using assms by blast
-  ultimately show "closedin (subtopology euclidean S) {x \<in> S. f x \<subseteq> U}"
+  ultimately show "closedin (top_of_set S) {x \<in> S. f x \<subseteq> U}"
     by (simp add: openin_closedin_eq)
 qed
 
 lemma open_map_iff_lower_hemicontinuous_preimage:
   assumes "f ` S \<subseteq> T"
-    shows "((\<forall>U. openin (subtopology euclidean S) U
-                 \<longrightarrow> openin (subtopology euclidean T) (f ` U)) \<longleftrightarrow>
-            (\<forall>U. closedin (subtopology euclidean S) U
-                 \<longrightarrow> closedin (subtopology euclidean T) {y \<in> T. {x. x \<in> S \<and> f x = y} \<subseteq> U}))"
+    shows "((\<forall>U. openin (top_of_set S) U
+                 \<longrightarrow> openin (top_of_set T) (f ` U)) \<longleftrightarrow>
+            (\<forall>U. closedin (top_of_set S) U
+                 \<longrightarrow> closedin (top_of_set T) {y \<in> T. {x. x \<in> S \<and> f x = y} \<subseteq> U}))"
           (is "?lhs = ?rhs")
 proof (intro iffI allI impI)
   fix U
-  assume * [rule_format]: ?lhs and "closedin (subtopology euclidean S) U"
-  then have "openin (subtopology euclidean S) (S - U)"
+  assume * [rule_format]: ?lhs and "closedin (top_of_set S) U"
+  then have "openin (top_of_set S) (S - U)"
     by (simp add: openin_diff)
-  then have "openin (subtopology euclidean T) (f ` (S - U))"
+  then have "openin (top_of_set T) (f ` (S - U))"
     using * [of "S-U"] by blast
   moreover have "T - (f ` (S - U)) = {y \<in> T. {x \<in> S. f x = y} \<subseteq> U}"
     using assms by blast
-  ultimately show "closedin (subtopology euclidean T) {y \<in> T. {x \<in> S. f x = y} \<subseteq> U}"
+  ultimately show "closedin (top_of_set T) {y \<in> T. {x \<in> S. f x = y} \<subseteq> U}"
     by (simp add: openin_closedin_eq)
 next
   fix U
-  assume * [rule_format]: ?rhs and opeSU: "openin (subtopology euclidean S) U"
-  then have "closedin (subtopology euclidean S) (S - U)"
+  assume * [rule_format]: ?rhs and opeSU: "openin (top_of_set S) U"
+  then have "closedin (top_of_set S) (S - U)"
     by (simp add: closedin_diff)
-  then have "closedin (subtopology euclidean T) {y \<in> T. {x \<in> S. f x = y} \<subseteq> S - U}"
+  then have "closedin (top_of_set T) {y \<in> T. {x \<in> S. f x = y} \<subseteq> S - U}"
     using * [of "S-U"] by blast
   moreover have "{y \<in> T. {x \<in> S. f x = y} \<subseteq> S - U} = T - (f ` U)"
     using assms openin_imp_subset [OF opeSU] by auto
-  ultimately show "openin (subtopology euclidean T) (f ` U)"
+  ultimately show "openin (top_of_set T) (f ` U)"
     using assms openin_imp_subset [OF opeSU] by (force simp: openin_closedin_eq)
 qed
 
 lemma closed_map_iff_upper_hemicontinuous_preimage:
   assumes "f ` S \<subseteq> T"
-    shows "((\<forall>U. closedin (subtopology euclidean S) U
-                 \<longrightarrow> closedin (subtopology euclidean T) (f ` U)) \<longleftrightarrow>
-            (\<forall>U. openin (subtopology euclidean S) U
-                 \<longrightarrow> openin (subtopology euclidean T) {y \<in> T. {x. x \<in> S \<and> f x = y} \<subseteq> U}))"
+    shows "((\<forall>U. closedin (top_of_set S) U
+                 \<longrightarrow> closedin (top_of_set T) (f ` U)) \<longleftrightarrow>
+            (\<forall>U. openin (top_of_set S) U
+                 \<longrightarrow> openin (top_of_set T) {y \<in> T. {x. x \<in> S \<and> f x = y} \<subseteq> U}))"
           (is "?lhs = ?rhs")
 proof (intro iffI allI impI)
   fix U
-  assume * [rule_format]: ?lhs and opeSU: "openin (subtopology euclidean S) U"
-  then have "closedin (subtopology euclidean S) (S - U)"
+  assume * [rule_format]: ?lhs and opeSU: "openin (top_of_set S) U"
+  then have "closedin (top_of_set S) (S - U)"
     by (simp add: closedin_diff)
-  then have "closedin (subtopology euclidean T) (f ` (S - U))"
+  then have "closedin (top_of_set T) (f ` (S - U))"
     using * [of "S-U"] by blast
   moreover have "f ` (S - U) = T -  {y \<in> T. {x. x \<in> S \<and> f x = y} \<subseteq> U}"
     using assms openin_imp_subset [OF opeSU] by auto
-  ultimately show "openin (subtopology euclidean T)  {y \<in> T. {x. x \<in> S \<and> f x = y} \<subseteq> U}"
+  ultimately show "openin (top_of_set T)  {y \<in> T. {x. x \<in> S \<and> f x = y} \<subseteq> U}"
     using assms openin_imp_subset [OF opeSU] by (force simp: openin_closedin_eq)
 next
   fix U
-  assume * [rule_format]: ?rhs and cloSU: "closedin (subtopology euclidean S) U"
-  then have "openin (subtopology euclidean S) (S - U)"
+  assume * [rule_format]: ?rhs and cloSU: "closedin (top_of_set S) U"
+  then have "openin (top_of_set S) (S - U)"
     by (simp add: openin_diff)
-  then have "openin (subtopology euclidean T) {y \<in> T. {x \<in> S. f x = y} \<subseteq> S - U}"
+  then have "openin (top_of_set T) {y \<in> T. {x \<in> S. f x = y} \<subseteq> S - U}"
     using * [of "S-U"] by blast
   moreover have "(f ` U) = T - {y \<in> T. {x \<in> S. f x = y} \<subseteq> S - U}"
     using assms closedin_imp_subset [OF cloSU]  by auto
-  ultimately show "closedin (subtopology euclidean T) (f ` U)"
+  ultimately show "closedin (top_of_set T) (f ` U)"
     by (simp add: openin_closedin_eq)
 qed
 
 proposition upper_lower_hemicontinuous_explicit:
   fixes T :: "('b::{real_normed_vector,heine_borel}) set"
   assumes fST: "\<And>x. x \<in> S \<Longrightarrow> f x \<subseteq> T"
-      and ope: "\<And>U. openin (subtopology euclidean T) U
-                     \<Longrightarrow> openin (subtopology euclidean S) {x \<in> S. f x \<subseteq> U}"
-      and clo: "\<And>U. closedin (subtopology euclidean T) U
-                     \<Longrightarrow> closedin (subtopology euclidean S) {x \<in> S. f x \<subseteq> U}"
+      and ope: "\<And>U. openin (top_of_set T) U
+                     \<Longrightarrow> openin (top_of_set S) {x \<in> S. f x \<subseteq> U}"
+      and clo: "\<And>U. closedin (top_of_set T) U
+                     \<Longrightarrow> closedin (top_of_set S) {x \<in> S. f x \<subseteq> U}"
       and "x \<in> S" "0 < e" and bofx: "bounded(f x)" and fx_ne: "f x \<noteq> {}"
   obtains d where "0 < d"
              "\<And>x'. \<lbrakk>x' \<in> S; dist x x' < d\<rbrakk>
                            \<Longrightarrow> (\<forall>y \<in> f x. \<exists>y'. y' \<in> f x' \<and> dist y y' < e) \<and>
                                (\<forall>y' \<in> f x'. \<exists>y. y \<in> f x \<and> dist y' y < e)"
 proof -
-  have "openin (subtopology euclidean T) (T \<inter> (\<Union>a\<in>f x. \<Union>b\<in>ball 0 e. {a + b}))"
+  have "openin (top_of_set T) (T \<inter> (\<Union>a\<in>f x. \<Union>b\<in>ball 0 e. {a + b}))"
     by (auto simp: open_sums openin_open_Int)
-  with ope have "openin (subtopology euclidean S)
+  with ope have "openin (top_of_set S)
                     {u \<in> S. f u \<subseteq> T \<inter> (\<Union>a\<in>f x. \<Union>b\<in>ball 0 e. {a + b})}" by blast
   with \<open>0 < e\<close> \<open>x \<in> S\<close> obtain d1 where "d1 > 0" and
          d1: "\<And>x'. \<lbrakk>x' \<in> S; dist x' x < d1\<rbrakk> \<Longrightarrow> f x' \<subseteq> T \<and> f x' \<subseteq> (\<Union>a \<in> f x. \<Union>b \<in> ball 0 e. {a + b})"
     by (force simp: openin_euclidean_subtopology_iff dest: fST)
-  have oo: "\<And>U. openin (subtopology euclidean T) U \<Longrightarrow>
-                 openin (subtopology euclidean S) {x \<in> S. f x \<inter> U \<noteq> {}}"
+  have oo: "\<And>U. openin (top_of_set T) U \<Longrightarrow>
+                 openin (top_of_set S) {x \<in> S. f x \<inter> U \<noteq> {}}"
     apply (rule lower_hemicontinuous [THEN iffD1, rule_format])
     using fST clo by auto
   have "compact (closure(f x))"
@@ -3806,9 +3811,9 @@
     by blast
   have xin: "x \<in> (\<Inter>a \<in> C. {x \<in> S. f x \<inter> T \<inter> ball a (e/2) \<noteq> {}})"
     using \<open>x \<in> S\<close> \<open>0 < e\<close> fST \<open>C \<subseteq> f x\<close> by force
-  have "openin (subtopology euclidean S) {x \<in> S. f x \<inter> (T \<inter> ball a (e/2)) \<noteq> {}}" for a
+  have "openin (top_of_set S) {x \<in> S. f x \<inter> (T \<inter> ball a (e/2)) \<noteq> {}}" for a
     by (simp add: openin_open_Int oo)
-  then have "openin (subtopology euclidean S) (\<Inter>a \<in> C. {x \<in> S. f x \<inter> T \<inter> ball a (e/2) \<noteq> {}})"
+  then have "openin (top_of_set S) (\<Inter>a \<in> C. {x \<in> S. f x \<inter> T \<inter> ball a (e/2) \<noteq> {}})"
     by (simp add: Int_assoc openin_INT2 [OF \<open>finite C\<close> \<open>C \<noteq> {}\<close>])
   with xin obtain d2 where "d2>0"
               and d2: "\<And>u v. \<lbrakk>u \<in> S; dist u x < d2; v \<in> C\<rbrakk> \<Longrightarrow> f u \<inter> T \<inter> ball v (e/2) \<noteq> {}"
@@ -4375,8 +4380,8 @@
 
 proposition Borsukian_open_Un:
   fixes S :: "'a::real_normed_vector set"
-  assumes opeS: "openin (subtopology euclidean (S \<union> T)) S"
-      and opeT: "openin (subtopology euclidean (S \<union> T)) T"
+  assumes opeS: "openin (top_of_set (S \<union> T)) S"
+      and opeT: "openin (top_of_set (S \<union> T)) T"
       and BS: "Borsukian S" and BT: "Borsukian T" and ST: "connected(S \<inter> T)"
     shows "Borsukian(S \<union> T)"
 proof (clarsimp simp add: Borsukian_continuous_logarithm)
@@ -4445,8 +4450,8 @@
 text\<open>The proof is a duplicate of that of \<open>Borsukian_open_Un\<close>.\<close>
 lemma Borsukian_closed_Un:
   fixes S :: "'a::real_normed_vector set"
-  assumes cloS: "closedin (subtopology euclidean (S \<union> T)) S"
-      and cloT: "closedin (subtopology euclidean (S \<union> T)) T"
+  assumes cloS: "closedin (top_of_set (S \<union> T)) S"
+      and cloT: "closedin (top_of_set (S \<union> T)) T"
       and BS: "Borsukian S" and BT: "Borsukian T" and ST: "connected(S \<inter> T)"
     shows "Borsukian(S \<union> T)"
 proof (clarsimp simp add: Borsukian_continuous_logarithm)
@@ -4586,8 +4591,8 @@
 lemma Borsukian_open_map_image_compact:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "Borsukian S" and contf: "continuous_on S f" and fim: "f ` S = T" and "compact S"
-      and ope: "\<And>U. openin (subtopology euclidean S) U
-                     \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
+      and ope: "\<And>U. openin (top_of_set S) U
+                     \<Longrightarrow> openin (top_of_set T) (f ` U)"
     shows "Borsukian T"
 proof (clarsimp simp add: Borsukian_continuous_logarithm_circle_real)
   fix g :: "'b \<Rightarrow> complex"
@@ -4636,12 +4641,12 @@
                \<Longrightarrow> (\<forall>v \<in> {z \<in> S. f z = y}. \<exists>v'. v' \<in> {z \<in> S. f z = x'} \<and> dist v v' < d) \<and>
                    (\<forall>v' \<in> {z \<in> S. f z = x'}. \<exists>v. v \<in> {z \<in> S. f z = y} \<and> dist v' v < d)"
     proof (rule upper_lower_hemicontinuous_explicit [of T "\<lambda>y. {z \<in> S. f z = y}" S])
-      show "\<And>U. openin (subtopology euclidean S) U
-                 \<Longrightarrow> openin (subtopology euclidean T) {x \<in> T. {z \<in> S. f z = x} \<subseteq> U}"
+      show "\<And>U. openin (top_of_set S) U
+                 \<Longrightarrow> openin (top_of_set T) {x \<in> T. {z \<in> S. f z = x} \<subseteq> U}"
         using continuous_imp_closed_map closed_map_iff_upper_hemicontinuous_preimage [OF fim [THEN equalityD1]]
         by (simp add: continuous_imp_closed_map \<open>compact S\<close> contf fim)
-      show "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow>
-                 closedin (subtopology euclidean T) {x \<in> T. {z \<in> S. f z = x} \<subseteq> U}"
+      show "\<And>U. closedin (top_of_set S) U \<Longrightarrow>
+                 closedin (top_of_set T) {x \<in> T. {z \<in> S. f z = x} \<subseteq> U}"
         using  ope open_map_iff_lower_hemicontinuous_preimage [OF fim [THEN equalityD1]]
         by meson
       show "bounded {z \<in> S. f z = y}"
@@ -4812,17 +4817,17 @@
 definition%important unicoherent where
   "unicoherent U \<equiv>
   \<forall>S T. connected S \<and> connected T \<and> S \<union> T = U \<and>
-        closedin (subtopology euclidean U) S \<and> closedin (subtopology euclidean U) T
+        closedin (top_of_set U) S \<and> closedin (top_of_set U) T
         \<longrightarrow> connected (S \<inter> T)"
 
 lemma unicoherentI [intro?]:
-  assumes "\<And>S T. \<lbrakk>connected S; connected T; U = S \<union> T; closedin (subtopology euclidean U) S; closedin (subtopology euclidean U) T\<rbrakk>
+  assumes "\<And>S T. \<lbrakk>connected S; connected T; U = S \<union> T; closedin (top_of_set U) S; closedin (top_of_set U) T\<rbrakk>
           \<Longrightarrow> connected (S \<inter> T)"
   shows "unicoherent U"
   using assms unfolding unicoherent_def by blast
 
 lemma unicoherentD:
-  assumes "unicoherent U" "connected S" "connected T" "U = S \<union> T" "closedin (subtopology euclidean U) S" "closedin (subtopology euclidean U) T"
+  assumes "unicoherent U" "connected S" "connected T" "U = S \<union> T" "closedin (top_of_set U) S" "closedin (top_of_set U) T"
   shows "connected (S \<inter> T)"
   using assms unfolding unicoherent_def by blast
 
@@ -4837,8 +4842,8 @@
   proof
     fix U V
     assume "connected U" "connected V" and T: "T = U \<union> V"
-      and cloU: "closedin (subtopology euclidean T) U"
-      and cloV: "closedin (subtopology euclidean T) V"
+      and cloU: "closedin (top_of_set T) U"
+      and cloV: "closedin (top_of_set T) V"
     have "f ` (g ` U \<inter> g ` V) \<subseteq> U" "f ` (g ` U \<inter> g ` V) \<subseteq> V"
       using gf fim T by auto (metis UnCI image_iff)+
     moreover have "U \<inter> V \<subseteq> f ` (g ` U \<inter> g ` V)"
@@ -4858,10 +4863,10 @@
           using T fim gfim by auto
         have hom: "homeomorphism T S g f"
           by (simp add: contf contg fim gf gfim homeomorphism_def)
-        have "closedin (subtopology euclidean T) U" "closedin (subtopology euclidean T) V"
+        have "closedin (top_of_set T) U" "closedin (top_of_set T) V"
           by (simp_all add: cloU cloV)
-        then show "closedin (subtopology euclidean S) (g ` U)"
-                  "closedin (subtopology euclidean S) (g ` V)"
+        then show "closedin (top_of_set S) (g ` U)"
+                  "closedin (top_of_set S) (g ` V)"
           by (blast intro: homeomorphism_imp_closed_map [OF hom])+
       qed
     qed
@@ -4894,16 +4899,16 @@
 proof clarify
   fix S T
   assume "connected S" "connected T" "U = S \<union> T"
-     and cloS: "closedin (subtopology euclidean (S \<union> T)) S"
-     and cloT: "closedin (subtopology euclidean (S \<union> T)) T"
+     and cloS: "closedin (top_of_set (S \<union> T)) S"
+     and cloT: "closedin (top_of_set (S \<union> T)) T"
   show "connected (S \<inter> T)"
     unfolding connected_closedin_eq
   proof clarify
     fix V W
-    assume "closedin (subtopology euclidean (S \<inter> T)) V"
-       and "closedin (subtopology euclidean (S \<inter> T)) W"
+    assume "closedin (top_of_set (S \<inter> T)) V"
+       and "closedin (top_of_set (S \<inter> T)) W"
        and VW: "V \<union> W = S \<inter> T" "V \<inter> W = {}" and "V \<noteq> {}" "W \<noteq> {}"
-    then have cloV: "closedin (subtopology euclidean U) V" and cloW: "closedin (subtopology euclidean U) W"
+    then have cloV: "closedin (top_of_set U) V" and cloW: "closedin (top_of_set U) W"
       using \<open>U = S \<union> T\<close> cloS cloT closedin_trans by blast+
     obtain q where contq: "continuous_on U q"
          and q01: "\<And>x. x \<in> U \<Longrightarrow> q x \<in> {0..1::real}"
@@ -5020,8 +5025,8 @@
 proof
   fix U V
   assume UV: "connected U" "connected V" "T = U \<union> V"
-     and cloU: "closedin (subtopology euclidean T) U"
-     and cloV: "closedin (subtopology euclidean T) V"
+     and cloU: "closedin (top_of_set T) U"
+     and cloV: "closedin (top_of_set T) V"
   moreover have "compact T"
     using \<open>compact S\<close> compact_continuous_image contf fim by blast
   ultimately have "closed U" "closed V"
@@ -5035,14 +5040,14 @@
       by (meson contf continuous_on_subset inf_le1)
     show "connected ?SUV"
     proof (rule unicoherentD [OF \<open>unicoherent S\<close>, of "S \<inter> f -` U" "S \<inter> f -` V"])
-      have "\<And>C. closedin (subtopology euclidean S) C \<Longrightarrow> closedin (subtopology euclidean T) (f ` C)"
+      have "\<And>C. closedin (top_of_set S) C \<Longrightarrow> closedin (top_of_set T) (f ` C)"
         by (metis \<open>compact S\<close> closed_subset closedin_compact closedin_imp_subset compact_continuous_image compact_imp_closed contf continuous_on_subset fim image_mono)
       then show "connected (S \<inter> f -` U)" "connected (S \<inter> f -` V)"
         using UV by (auto simp: conn intro: connected_closed_monotone_preimage [OF contf fim])
       show "S = (S \<inter> f -` U) \<union> (S \<inter> f -` V)"
         using UV fim by blast
-      show "closedin (subtopology euclidean S) (S \<inter> f -` U)"
-            "closedin (subtopology euclidean S) (S \<inter> f -` V)"
+      show "closedin (top_of_set S) (S \<inter> f -` U)"
+            "closedin (top_of_set S) (S \<inter> f -` V)"
         by (auto simp: continuous_on_imp_closedin cloU cloV contf fim)
     qed
   qed
@@ -5348,7 +5353,7 @@
     obtain g where contg: "continuous_on UNIV g" and gim: "g ` UNIV \<subseteq> -{0}"
                    and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
     proof (rule Borsuk_homotopy_extension_homotopic [OF _ _ continuous_on_const _ homotopic_with_symD [OF a]])
-      show "closedin (subtopology euclidean UNIV) S"
+      show "closedin (top_of_set UNIV) S"
         using assms by auto
       show "range (\<lambda>t. a) \<subseteq> - {0}"
         using a homotopic_with_imp_subset2 False by blast
@@ -5396,8 +5401,8 @@
 lemma inessential_on_clopen_Union:
   fixes \<F> :: "'a::euclidean_space set set"
   assumes T: "path_connected T"
-      and "\<And>S. S \<in> \<F> \<Longrightarrow> closedin (subtopology euclidean (\<Union>\<F>)) S"
-      and "\<And>S. S \<in> \<F> \<Longrightarrow> openin (subtopology euclidean (\<Union>\<F>)) S"
+      and "\<And>S. S \<in> \<F> \<Longrightarrow> closedin (top_of_set (\<Union>\<F>)) S"
+      and "\<And>S. S \<in> \<F> \<Longrightarrow> openin (top_of_set (\<Union>\<F>)) S"
       and hom: "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. a)"
   obtains a where "homotopic_with (\<lambda>x. True) (\<Union>\<F>) T f (\<lambda>x. a)"
 proof (cases "\<Union>\<F> = {}")
@@ -5408,16 +5413,16 @@
   case False
   then obtain C where "C \<in> \<F>" "C \<noteq> {}"
     by blast
-  then obtain a where clo: "closedin (subtopology euclidean (\<Union>\<F>)) C"
-    and ope: "openin (subtopology euclidean (\<Union>\<F>)) C"
+  then obtain a where clo: "closedin (top_of_set (\<Union>\<F>)) C"
+    and ope: "openin (top_of_set (\<Union>\<F>)) C"
     and "homotopic_with (\<lambda>x. True) C T f (\<lambda>x. a)"
     using assms by blast
   with \<open>C \<noteq> {}\<close> have "f ` C \<subseteq> T" "a \<in> T"
     using homotopic_with_imp_subset1 homotopic_with_imp_subset2 by blast+
   have "homotopic_with (\<lambda>x. True) (\<Union>\<F>) T f (\<lambda>x. a)"
   proof (rule homotopic_on_clopen_Union)
-    show "\<And>S. S \<in> \<F> \<Longrightarrow> closedin (subtopology euclidean (\<Union>\<F>)) S"
-         "\<And>S. S \<in> \<F> \<Longrightarrow> openin (subtopology euclidean (\<Union>\<F>)) S"
+    show "\<And>S. S \<in> \<F> \<Longrightarrow> closedin (top_of_set (\<Union>\<F>)) S"
+         "\<And>S. S \<in> \<F> \<Longrightarrow> openin (top_of_set (\<Union>\<F>)) S"
       by (simp_all add: assms)
     show "homotopic_with (\<lambda>x. True) S T f (\<lambda>x. a)" if "S \<in> \<F>" for S
     proof (cases "S = {}")
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -4415,9 +4415,9 @@
     then show "\<exists>e>0. ball x e \<subseteq> (S \<inter> f -` {f x})"
       using \<open>0 < e\<close> by blast
   qed
-  then have "openin (subtopology euclidean S) (S \<inter> f -` {y})"
+  then have "openin (top_of_set S) (S \<inter> f -` {y})"
     by (auto intro!: open_openin_trans[OF \<open>open S\<close>] simp: open_contains_ball)
-  moreover have "closedin (subtopology euclidean S) (S \<inter> f -` {y})"
+  moreover have "closedin (top_of_set S) (S \<inter> f -` {y})"
     by (force intro!: continuous_closedin_preimage [OF contf])
   ultimately have "(S \<inter> f -` {y}) = {} \<or> (S \<inter> f -` {y}) = S"
     using \<open>connected S\<close> by (simp add: connected_clopen)
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Homeomorphism.thy
--- a/src/HOL/Analysis/Homeomorphism.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Homeomorphism.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -916,7 +916,7 @@
   fixes S :: "'m::euclidean_space set"
   assumes "aff_dim S < DIM('n)"
   obtains U and T :: "'n::euclidean_space set"
-     where "convex U" "U \<noteq> {}" "closedin (subtopology euclidean U) T"
+     where "convex U" "U \<noteq> {}" "closedin (top_of_set U) T"
            "S homeomorphic T"
 proof (cases "S = {}")
   case True then show ?thesis
@@ -1051,7 +1051,7 @@
 lemma locally_compact_closedin_open:
     fixes S :: "'a :: metric_space set"
     assumes "locally compact S"
-    obtains T where "open T" "closedin (subtopology euclidean T) S"
+    obtains T where "open T" "closedin (top_of_set T) S"
   by (metis locally_compact_open_Int_closure [OF assms] closed_closure closedin_closed_Int)
 
 
@@ -1079,7 +1079,7 @@
       by (simp add: Ucomp setdist_eq_0_sing_1)
     then have homU: "homeomorphism U (f`U) f fst"
       by (auto simp: f_def homeomorphism_def image_iff continuous_intros)
-    have cloS: "closedin (subtopology euclidean U) S"
+    have cloS: "closedin (top_of_set U) S"
       by (metis US closed_closure closedin_closed_Int)
     have cont: "isCont ((\<lambda>x. setdist {x} (- U)) o fst) z" for z :: "'a \<times> 'b"
       by (rule continuous_at_compose continuous_intros continuous_at_setdist)+
@@ -1275,9 +1275,9 @@
   where
   "covering_space c p S \<equiv>
        continuous_on c p \<and> p ` c = S \<and>
-       (\<forall>x \<in> S. \<exists>T. x \<in> T \<and> openin (subtopology euclidean S) T \<and>
+       (\<forall>x \<in> S. \<exists>T. x \<in> T \<and> openin (top_of_set S) T \<and>
                     (\<exists>v. \<Union>v = c \<inter> p -` T \<and>
-                        (\<forall>u \<in> v. openin (subtopology euclidean c) u) \<and>
+                        (\<forall>u \<in> v. openin (top_of_set c) u) \<and>
                         pairwise disjnt v \<and>
                         (\<forall>u \<in> v. \<exists>q. homeomorphism u T p q)))"
 
@@ -1295,8 +1295,8 @@
 
 lemma covering_space_local_homeomorphism:
   assumes "covering_space c p S" "x \<in> c"
-  obtains T u q where "x \<in> T" "openin (subtopology euclidean c) T"
-                      "p x \<in> u" "openin (subtopology euclidean S) u"
+  obtains T u q where "x \<in> T" "openin (top_of_set c) T"
+                      "p x \<in> u" "openin (top_of_set S) u"
                       "homeomorphism T u p q"
 using assms
 apply (simp add: covering_space_def, clarify)
@@ -1307,8 +1307,8 @@
 lemma covering_space_local_homeomorphism_alt:
   assumes p: "covering_space c p S" and "y \<in> S"
   obtains x T U q where "p x = y"
-                        "x \<in> T" "openin (subtopology euclidean c) T"
-                        "y \<in> U" "openin (subtopology euclidean S) U"
+                        "x \<in> T" "openin (top_of_set c) T"
+                        "y \<in> U" "openin (top_of_set S) U"
                           "homeomorphism T U p q"
 proof -
   obtain x where "p x = y" "x \<in> c"
@@ -1320,26 +1320,26 @@
 
 proposition covering_space_open_map:
   fixes S :: "'a :: metric_space set" and T :: "'b :: metric_space set"
-  assumes p: "covering_space c p S" and T: "openin (subtopology euclidean c) T"
-    shows "openin (subtopology euclidean S) (p ` T)"
+  assumes p: "covering_space c p S" and T: "openin (top_of_set c) T"
+    shows "openin (top_of_set S) (p ` T)"
 proof -
   have pce: "p ` c = S"
    and covs:
        "\<And>x. x \<in> S \<Longrightarrow>
-            \<exists>X VS. x \<in> X \<and> openin (subtopology euclidean S) X \<and>
+            \<exists>X VS. x \<in> X \<and> openin (top_of_set S) X \<and>
                   \<Union>VS = c \<inter> p -` X \<and>
-                  (\<forall>u \<in> VS. openin (subtopology euclidean c) u) \<and>
+                  (\<forall>u \<in> VS. openin (top_of_set c) u) \<and>
                   pairwise disjnt VS \<and>
                   (\<forall>u \<in> VS. \<exists>q. homeomorphism u X p q)"
     using p by (auto simp: covering_space_def)
   have "T \<subseteq> c"  by (metis openin_euclidean_subtopology_iff T)
-  have "\<exists>X. openin (subtopology euclidean S) X \<and> y \<in> X \<and> X \<subseteq> p ` T"
+  have "\<exists>X. openin (top_of_set S) X \<and> y \<in> X \<and> X \<subseteq> p ` T"
           if "y \<in> p ` T" for y
   proof -
     have "y \<in> S" using \<open>T \<subseteq> c\<close> pce that by blast
-    obtain U VS where "y \<in> U" and U: "openin (subtopology euclidean S) U"
+    obtain U VS where "y \<in> U" and U: "openin (top_of_set S) U"
                   and VS: "\<Union>VS = c \<inter> p -` U"
-                  and openVS: "\<forall>V \<in> VS. openin (subtopology euclidean c) V"
+                  and openVS: "\<forall>V \<in> VS. openin (top_of_set c) V"
                   and homVS: "\<And>V. V \<in> VS \<Longrightarrow> \<exists>q. homeomorphism V U p q"
       using covs [OF \<open>y \<in> S\<close>] by auto
     obtain x where "x \<in> c" "p x \<in> U" "x \<in> T" "p x = y"
@@ -1349,14 +1349,14 @@
     then obtain q where q: "homeomorphism V U p q" using homVS by blast
     then have ptV: "p ` (T \<inter> V) = U \<inter> q -` (T \<inter> V)"
       using VS \<open>V \<in> VS\<close> by (auto simp: homeomorphism_def)
-    have ocv: "openin (subtopology euclidean c) V"
+    have ocv: "openin (top_of_set c) V"
       by (simp add: \<open>V \<in> VS\<close> openVS)
-    have "openin (subtopology euclidean U) (U \<inter> q -` (T \<inter> V))"
+    have "openin (top_of_set U) (U \<inter> q -` (T \<inter> V))"
       apply (rule continuous_on_open [THEN iffD1, rule_format])
        using homeomorphism_def q apply blast
       using openin_subtopology_Int_subset [of c] q T unfolding homeomorphism_def
       by (metis inf.absorb_iff2 Int_commute ocv openin_euclidean_subtopology_iff)
-    then have os: "openin (subtopology euclidean S) (U \<inter> q -` (T \<inter> V))"
+    then have os: "openin (top_of_set S) (U \<inter> q -` (T \<inter> V))"
       using openin_trans [of U] by (simp add: Collect_conj_eq U)
     show ?thesis
       apply (rule_tac x = "p ` (T \<inter> V)" in exI)
@@ -1386,13 +1386,13 @@
   have "connected U" by (rule in_components_connected [OF u_compt])
   have contu: "continuous_on U g1" "continuous_on U g2"
        using \<open>U \<subseteq> T\<close> continuous_on_subset g1 g2 by blast+
-  have o12: "openin (subtopology euclidean U) G12"
+  have o12: "openin (top_of_set U) G12"
   unfolding G12_def
   proof (subst openin_subopen, clarify)
     fix z
     assume z: "z \<in> U" "g1 z - g2 z = 0"
-    obtain v w q where "g1 z \<in> v" and ocv: "openin (subtopology euclidean c) v"
-                   and "p (g1 z) \<in> w" and osw: "openin (subtopology euclidean S) w"
+    obtain v w q where "g1 z \<in> v" and ocv: "openin (top_of_set c) v"
+                   and "p (g1 z) \<in> w" and osw: "openin (top_of_set S) w"
                    and hom: "homeomorphism v w p q"
       apply (rule_tac x = "g1 z" in covering_space_local_homeomorphism [OF cov])
        using \<open>U \<subseteq> T\<close> \<open>z \<in> U\<close> g1(2) apply blast+
@@ -1400,15 +1400,15 @@
     have "g2 z \<in> v" using \<open>g1 z \<in> v\<close> z by auto
     have gg: "U \<inter> g -` v = U \<inter> g -` (v \<inter> g ` U)" for g
       by auto
-    have "openin (subtopology euclidean (g1 ` U)) (v \<inter> g1 ` U)"
+    have "openin (top_of_set (g1 ` U)) (v \<inter> g1 ` U)"
       using ocv \<open>U \<subseteq> T\<close> g1 by (fastforce simp add: openin_open)
-    then have 1: "openin (subtopology euclidean U) (U \<inter> g1 -` v)"
+    then have 1: "openin (top_of_set U) (U \<inter> g1 -` v)"
       unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1, rule_format])
-    have "openin (subtopology euclidean (g2 ` U)) (v \<inter> g2 ` U)"
+    have "openin (top_of_set (g2 ` U)) (v \<inter> g2 ` U)"
       using ocv \<open>U \<subseteq> T\<close> g2 by (fastforce simp add: openin_open)
-    then have 2: "openin (subtopology euclidean U) (U \<inter> g2 -` v)"
+    then have 2: "openin (top_of_set U) (U \<inter> g2 -` v)"
       unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1, rule_format])
-    show "\<exists>T. openin (subtopology euclidean U) T \<and> z \<in> T \<and> T \<subseteq> {z \<in> U. g1 z - g2 z = 0}"
+    show "\<exists>T. openin (top_of_set U) T \<and> z \<in> T \<and> T \<subseteq> {z \<in> U. g1 z - g2 z = 0}"
       using z
       apply (rule_tac x = "(U \<inter> g1 -` v) \<inter> (U \<inter> g2 -` v)" in exI)
       apply (intro conjI)
@@ -1417,7 +1417,7 @@
       apply (metis \<open>U \<subseteq> T\<close> subsetD eq_iff_diff_eq_0 fg1 fg2 hom homeomorphism_def)
       done
   qed
-  have c12: "closedin (subtopology euclidean U) G12"
+  have c12: "closedin (top_of_set U) G12"
     unfolding G12_def
     by (intro continuous_intros continuous_closedin_preimage_constant contu)
   have "G12 = {} \<or> G12 = U"
@@ -1468,37 +1468,37 @@
   show ?rhs
   proof (rule locallyI)
     fix V x
-    assume V: "openin (subtopology euclidean C) V" and "x \<in> V"
+    assume V: "openin (top_of_set C) V" and "x \<in> V"
     have "p x \<in> p ` C"
       by (metis IntE V \<open>x \<in> V\<close> imageI openin_open)
     then obtain T \<V> where "p x \<in> T"
-                      and opeT: "openin (subtopology euclidean S) T"
+                      and opeT: "openin (top_of_set S) T"
                       and veq: "\<Union>\<V> = C \<inter> p -` T"
-                      and ope: "\<forall>U\<in>\<V>. openin (subtopology euclidean C) U"
+                      and ope: "\<forall>U\<in>\<V>. openin (top_of_set C) U"
                       and hom: "\<forall>U\<in>\<V>. \<exists>q. homeomorphism U T p q"
       using cov unfolding covering_space_def by (blast intro: that)
     have "x \<in> \<Union>\<V>"
       using V veq \<open>p x \<in> T\<close> \<open>x \<in> V\<close> openin_imp_subset by fastforce
     then obtain U where "x \<in> U" "U \<in> \<V>"
       by blast
-    then obtain q where opeU: "openin (subtopology euclidean C) U" and q: "homeomorphism U T p q"
+    then obtain q where opeU: "openin (top_of_set C) U" and q: "homeomorphism U T p q"
       using ope hom by blast
-    with V have "openin (subtopology euclidean C) (U \<inter> V)"
+    with V have "openin (top_of_set C) (U \<inter> V)"
       by blast
-    then have UV: "openin (subtopology euclidean S) (p ` (U \<inter> V))"
+    then have UV: "openin (top_of_set S) (p ` (U \<inter> V))"
       using cov covering_space_open_map by blast
-    obtain W W' where opeW: "openin (subtopology euclidean S) W" and "\<psi> W'" "p x \<in> W" "W \<subseteq> W'" and W'sub: "W' \<subseteq> p ` (U \<inter> V)"
+    obtain W W' where opeW: "openin (top_of_set S) W" and "\<psi> W'" "p x \<in> W" "W \<subseteq> W'" and W'sub: "W' \<subseteq> p ` (U \<inter> V)"
       using locallyE [OF L UV] \<open>x \<in> U\<close> \<open>x \<in> V\<close> by blast
     then have "W \<subseteq> T"
       by (metis Int_lower1 q homeomorphism_image1 image_Int_subset order_trans)
-    show "\<exists>U Z. openin (subtopology euclidean C) U \<and>
+    show "\<exists>U Z. openin (top_of_set C) U \<and>
                  \<phi> Z \<and> x \<in> U \<and> U \<subseteq> Z \<and> Z \<subseteq> V"
     proof (intro exI conjI)
-      have "openin (subtopology euclidean T) W"
+      have "openin (top_of_set T) W"
         by (meson opeW opeT openin_imp_subset openin_subset_trans \<open>W \<subseteq> T\<close>)
-      then have "openin (subtopology euclidean U) (q ` W)"
+      then have "openin (top_of_set U) (q ` W)"
         by (meson homeomorphism_imp_open_map homeomorphism_symD q)
-      then show "openin (subtopology euclidean C) (q ` W)"
+      then show "openin (top_of_set C) (q ` W)"
         using opeU openin_trans by blast
       show "\<phi> (q ` W')"
         by (metis (mono_tags, lifting) Int_subset_iff UV W'sub \<open>\<psi> W'\<close> continuous_on_subset dual_order.trans homeomorphism_def image_Int_subset openin_imp_subset q qim)
@@ -1575,18 +1575,18 @@
                     "\<And>y. y \<in> U \<Longrightarrow> k(0, y) = f y"
                     "\<And>z. z \<in> {0..1} \<times> U \<Longrightarrow> h z = p(k z)"
 proof -
-  have "\<exists>V k. openin (subtopology euclidean U) V \<and> y \<in> V \<and>
+  have "\<exists>V k. openin (top_of_set U) V \<and> y \<in> V \<and>
               continuous_on ({0..1} \<times> V) k \<and> k ` ({0..1} \<times> V) \<subseteq> C \<and>
               (\<forall>z \<in> V. k(0, z) = f z) \<and> (\<forall>z \<in> {0..1} \<times> V. h z = p(k z))"
         if "y \<in> U" for y
   proof -
-    obtain UU where UU: "\<And>s. s \<in> S \<Longrightarrow> s \<in> (UU s) \<and> openin (subtopology euclidean S) (UU s) \<and>
+    obtain UU where UU: "\<And>s. s \<in> S \<Longrightarrow> s \<in> (UU s) \<and> openin (top_of_set S) (UU s) \<and>
                                         (\<exists>\<V>. \<Union>\<V> = C \<inter> p -` UU s \<and>
-                                            (\<forall>U \<in> \<V>. openin (subtopology euclidean C) U) \<and>
+                                            (\<forall>U \<in> \<V>. openin (top_of_set C) U) \<and>
                                             pairwise disjnt \<V> \<and>
                                             (\<forall>U \<in> \<V>. \<exists>q. homeomorphism U (UU s) p q))"
       using cov unfolding covering_space_def by (metis (mono_tags))
-    then have ope: "\<And>s. s \<in> S \<Longrightarrow> s \<in> (UU s) \<and> openin (subtopology euclidean S) (UU s)"
+    then have ope: "\<And>s. s \<in> S \<Longrightarrow> s \<in> (UU s) \<and> openin (top_of_set S) (UU s)"
       by blast
     have "\<exists>k n i. open k \<and> open n \<and>
                   t \<in> k \<and> y \<in> n \<and> i \<in> S \<and> h ` (({0..1} \<inter> k) \<times> (U \<inter> n)) \<subseteq> UU i" if "t \<in> {0..1}" for t
@@ -1595,7 +1595,7 @@
         using \<open>y \<in> U\<close> him that by blast
       then have "(t,y) \<in> ({0..1} \<times> U) \<inter> h -` UU(h(t, y))"
         using \<open>y \<in> U\<close> \<open>t \<in> {0..1}\<close>  by (auto simp: ope)
-      moreover have ope_01U: "openin (subtopology euclidean ({0..1} \<times> U)) (({0..1} \<times> U) \<inter> h -` UU(h(t, y)))"
+      moreover have ope_01U: "openin (top_of_set ({0..1} \<times> U)) (({0..1} \<times> U) \<inter> h -` UU(h(t, y)))"
         using hinS ope continuous_on_open_gen [OF him] conth by blast
       ultimately obtain V W where opeV: "open V" and "t \<in> {0..1} \<inter> V" "t \<in> {0..1} \<inter> V"
                               and opeW: "open W" and "y \<in> U" "y \<in> W"
@@ -1641,7 +1641,7 @@
       using reals_Archimedean2 by blast
     then have "N > 0"
       using \<open>0 < \<delta>\<close> order.asym by force
-    have *: "\<exists>V k. openin (subtopology euclidean U) V \<and> y \<in> V \<and>
+    have *: "\<exists>V k. openin (top_of_set U) V \<and> y \<in> V \<and>
                    continuous_on ({0..of_nat n / N} \<times> V) k \<and>
                    k ` ({0..of_nat n / N} \<times> V) \<subseteq> C \<and>
                    (\<forall>z\<in>V. k (0, z) = f z) \<and>
@@ -1657,7 +1657,7 @@
         done
     next
       case (Suc n)
-      then obtain V k where opeUV: "openin (subtopology euclidean U) V"
+      then obtain V k where opeUV: "openin (top_of_set U) V"
                         and "y \<in> V"
                         and contk: "continuous_on ({0..real n / real N} \<times> V) k"
                         and kim: "k ` ({0..real n / real N} \<times> V) \<subseteq> C"
@@ -1676,7 +1676,7 @@
       have t01: "t \<in> {0..1}"
         using \<open>t \<in> tk\<close> \<open>tk \<subseteq> {0..1}\<close> by blast
       obtain \<V> where \<V>: "\<Union>\<V> = C \<inter> p -` UU (X t)"
-                 and opeC: "\<And>U. U \<in> \<V> \<Longrightarrow> openin (subtopology euclidean C) U"
+                 and opeC: "\<And>U. U \<in> \<V> \<Longrightarrow> openin (top_of_set C) U"
                  and "pairwise disjnt \<V>"
                  and homuu: "\<And>U. U \<in> \<V> \<Longrightarrow> \<exists>q. homeomorphism U (UU (X t)) p q"
         using inUS [OF t01] UU by meson
@@ -1703,24 +1703,24 @@
         by blast
       then obtain W where W: "k (real n / real N, y) \<in> W" and "W \<in> \<V>"
         by blast
-      then obtain p' where opeC': "openin (subtopology euclidean C) W"
+      then obtain p' where opeC': "openin (top_of_set C) W"
                        and hom': "homeomorphism W (UU (X t)) p p'"
         using homuu opeC by blast
       then have "W \<subseteq> C"
         using openin_imp_subset by blast
       define W' where "W' = UU(X t)"
-      have opeVW: "openin (subtopology euclidean V) (V \<inter> (k \<circ> Pair (n / N)) -` W)"
+      have opeVW: "openin (top_of_set V) (V \<inter> (k \<circ> Pair (n / N)) -` W)"
         apply (rule continuous_openin_preimage [OF _ _ opeC'])
          apply (intro continuous_intros continuous_on_subset [OF contk])
         using kim apply (auto simp: \<open>y \<in> V\<close> W)
         done
-      obtain N' where opeUN': "openin (subtopology euclidean U) N'"
+      obtain N' where opeUN': "openin (top_of_set U) N'"
                   and "y \<in> N'" and kimw: "k ` ({(real n / real N)} \<times> N') \<subseteq> W"
         apply (rule_tac N' = "(V \<inter> (k \<circ> Pair (n / N)) -` W)" in that)
         apply (fastforce simp:  \<open>y \<in> V\<close> W intro!: openin_trans [OF opeVW opeUV])+
         done
-      obtain Q Q' where opeUQ: "openin (subtopology euclidean U) Q"
-                    and cloUQ': "closedin (subtopology euclidean U) Q'"
+      obtain Q Q' where opeUQ: "openin (top_of_set U) Q"
+                    and cloUQ': "closedin (top_of_set U) Q'"
                     and "y \<in> Q" "Q \<subseteq> Q'"
                     and Q': "Q' \<subseteq> (U \<inter> NN(t)) \<inter> N' \<inter> V"
       proof -
@@ -1732,9 +1732,9 @@
           by (metis Int_iff \<open>N' = U \<inter> VX\<close> \<open>V = U \<inter> VO\<close> \<open>y \<in> N'\<close> \<open>y \<in> V\<close> inUS open_contains_cball t01)
         show ?thesis
         proof
-          show "openin (subtopology euclidean U) (U \<inter> ball y e)"
+          show "openin (top_of_set U) (U \<inter> ball y e)"
             by blast
-          show "closedin (subtopology euclidean U) (U \<inter> cball y e)"
+          show "closedin (top_of_set U) (U \<inter> cball y e)"
             using e by (auto simp: closedin_closed)
         qed (use \<open>y \<in> U\<close> \<open>e > 0\<close> VO VX e in auto)
       qed
@@ -1749,13 +1749,13 @@
         (\<lambda>x. if x \<in> {0..real n / real N} \<times> Q' then k x else (p' \<circ> h) x)"
         unfolding neqQ' [symmetric]
       proof (rule continuous_on_cases_local, simp_all add: neqQ' del: comp_apply)
-        show "closedin (subtopology euclidean ({0..(1 + real n) / real N} \<times> Q'))
+        show "closedin (top_of_set ({0..(1 + real n) / real N} \<times> Q'))
                        ({0..real n / real N} \<times> Q')"
           apply (simp add: closedin_closed)
           apply (rule_tac x="{0 .. real n / real N} \<times> UNIV" in exI)
           using n_div_N_in apply (auto simp: closed_Times)
           done
-        show "closedin (subtopology euclidean ({0..(1 + real n) / real N} \<times> Q'))
+        show "closedin (top_of_set ({0..(1 + real n) / real N} \<times> Q'))
                        ({real n / real N..(1 + real n) / real N} \<times> Q')"
           apply (simp add: closedin_closed)
           apply (rule_tac x="{real n / real N .. (1 + real n) / real N} \<times> UNIV" in exI)
@@ -1845,7 +1845,7 @@
     show ?thesis
       using*[OF order_refl] N \<open>0 < \<delta>\<close> by (simp add: split: if_split_asm)
   qed
-  then obtain V fs where opeV: "\<And>y. y \<in> U \<Longrightarrow> openin (subtopology euclidean U) (V y)"
+  then obtain V fs where opeV: "\<And>y. y \<in> U \<Longrightarrow> openin (top_of_set U) (V y)"
           and V: "\<And>y. y \<in> U \<Longrightarrow> y \<in> V y"
           and contfs: "\<And>y. y \<in> U \<Longrightarrow> continuous_on ({0..1} \<times> V y) (fs y)"
           and *: "\<And>y. y \<in> U \<Longrightarrow> (fs y) ` ({0..1} \<times> V y) \<subseteq> C \<and>
@@ -1857,14 +1857,17 @@
   obtain k where contk: "continuous_on ({0..1} \<times> U) k"
              and k: "\<And>x i. \<lbrakk>i \<in> U; x \<in> {0..1} \<times> U \<inter> {0..1} \<times> V i\<rbrakk> \<Longrightarrow> k x = fs i x"
   proof (rule pasting_lemma_exists)
-    show "{0..1} \<times> U \<subseteq> (\<Union>i\<in>U. {0..1} \<times> V i)"
-      apply auto
-      using V by blast
-    show "\<And>i. i \<in> U \<Longrightarrow> openin (subtopology euclidean ({0..1} \<times> U)) ({0..1} \<times> V i)"
+    let ?X = "top_of_set ({0..1::real} \<times> U)"
+    show "topspace ?X \<subseteq> (\<Union>i\<in>U. {0..1} \<times> V i)"
+      using V by force
+    show "\<And>i. i \<in> U \<Longrightarrow> openin (top_of_set ({0..1} \<times> U)) ({0..1} \<times> V i)"
       by (simp add: opeV openin_Times)
-    show "\<And>i. i \<in> U \<Longrightarrow> continuous_on ({0..1} \<times> V i) (fs i)"
-      using contfs by blast
-    show "fs i x = fs j x"  if "i \<in> U" "j \<in> U" and x: "x \<in> {0..1} \<times> U \<inter> {0..1} \<times> V i \<inter> {0..1} \<times> V j"
+    show "\<And>i. i \<in> U \<Longrightarrow> continuous_map
+              (subtopology (top_of_set ({0..1} \<times> U)) ({0..1} \<times> V i)) euclidean (fs i)"
+      using contfs
+      apply simp
+      by (metis continuous_map_iff_continuous continuous_on_subset openin_imp_subset openin_subtopology_self subtopology_subtopology)
+    show "fs i x = fs j x"  if "i \<in> U" "j \<in> U" and x: "x \<in> topspace ?X \<inter> {0..1} \<times> V i \<inter> {0..1} \<times> V j"
          for i j x
     proof -
       obtain u y where "x = (u, y)" "y \<in> V i" "y \<in> V j" "0 \<le> u" "u \<le> 1"
@@ -1900,7 +1903,7 @@
           using \<open>x = (u, y)\<close> x by blast
       qed
     qed
-  qed blast
+  qed force
   show ?thesis
   proof
     show "k ` ({0..1} \<times> U) \<subseteq> C"
@@ -2282,17 +2285,17 @@
       using l [of "linepath z z" z "linepath a a"] by (auto simp: assms)
     show LC: "l ` U \<subseteq> C"
       by (clarify dest!: *) (metis (full_types) l pathfinish_in_path_image subsetCE)
-    have "\<exists>T. openin (subtopology euclidean U) T \<and> y \<in> T \<and> T \<subseteq> U \<inter> l -` X"
-         if X: "openin (subtopology euclidean C) X" and "y \<in> U" "l y \<in> X" for X y
+    have "\<exists>T. openin (top_of_set U) T \<and> y \<in> T \<and> T \<subseteq> U \<inter> l -` X"
+         if X: "openin (top_of_set C) X" and "y \<in> U" "l y \<in> X" for X y
     proof -
       have "X \<subseteq> C"
         using X openin_euclidean_subtopology_iff by blast
       have "f y \<in> S"
         using fim \<open>y \<in> U\<close> by blast
       then obtain W \<V>
-              where WV: "f y \<in> W \<and> openin (subtopology euclidean S) W \<and>
+              where WV: "f y \<in> W \<and> openin (top_of_set S) W \<and>
                          (\<Union>\<V> = C \<inter> p -` W \<and>
-                          (\<forall>U \<in> \<V>. openin (subtopology euclidean C) U) \<and>
+                          (\<forall>U \<in> \<V>. openin (top_of_set C) U) \<and>
                           pairwise disjnt \<V> \<and>
                           (\<forall>U \<in> \<V>. \<exists>q. homeomorphism U W p q))"
         using cov by (force simp: covering_space_def)
@@ -2300,15 +2303,15 @@
         using \<open>X \<subseteq> C\<close> pleq that by auto
       then obtain W' where "l y \<in> W'" and "W' \<in> \<V>"
         by blast
-      with WV obtain p' where opeCW': "openin (subtopology euclidean C) W'"
+      with WV obtain p' where opeCW': "openin (top_of_set C) W'"
                           and homUW': "homeomorphism W' W p p'"
         by blast
       then have contp': "continuous_on W p'" and p'im: "p' ` W \<subseteq> W'"
         using homUW' homeomorphism_image2 homeomorphism_cont2 by fastforce+
       obtain V where "y \<in> V" "y \<in> U" and fimW: "f ` V \<subseteq> W" "V \<subseteq> U"
-                 and "path_connected V" and opeUV: "openin (subtopology euclidean U) V"
+                 and "path_connected V" and opeUV: "openin (top_of_set U) V"
       proof -
-        have "openin (subtopology euclidean U) (U \<inter> f -` W)"
+        have "openin (top_of_set U) (U \<inter> f -` W)"
           using WV contf continuous_on_open_gen fim by auto
         then show ?thesis
           using U WV
@@ -2324,16 +2327,16 @@
         using homUW' homeomorphism_image2 by fastforce
       show ?thesis
       proof (intro exI conjI)
-        have "openin (subtopology euclidean S) (W \<inter> p' -` (W' \<inter> X))"
+        have "openin (top_of_set S) (W \<inter> p' -` (W' \<inter> X))"
         proof (rule openin_trans)
-          show "openin (subtopology euclidean W) (W \<inter> p' -` (W' \<inter> X))"
+          show "openin (top_of_set W) (W \<inter> p' -` (W' \<inter> X))"
             apply (rule continuous_openin_preimage [OF contp' p'im])
             using X \<open>W' \<subseteq> C\<close> apply (auto simp: openin_open)
             done
-          show "openin (subtopology euclidean S) W"
+          show "openin (top_of_set S) W"
             using WV by blast
         qed
-        then show "openin (subtopology euclidean U) (V \<inter> (U \<inter> (f -` (W \<inter> (p' -` (W' \<inter> X))))))"
+        then show "openin (top_of_set U) (V \<inter> (U \<inter> (f -` (W \<inter> (p' -` (W' \<inter> X))))))"
           by (blast intro: opeUV openin_subtopology_self continuous_openin_preimage [OF contf fim])
          have "p' (f y) \<in> X"
           using \<open>l y \<in> W'\<close> homeomorphism_apply1 [OF homUW'] pleq \<open>y \<in> U\<close> \<open>l y \<in> X\<close> by fastforce
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Homotopy.thy
--- a/src/HOL/Analysis/Homotopy.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Homotopy.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -258,12 +258,12 @@
     assumes "homotopic_with P X Y f g" and "homotopic_with P X Y g h"
       shows "homotopic_with P X Y f h"
 proof -
-  have clo1: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({0..1/2::real} \<times> X)"
+  have clo1: "closedin (top_of_set ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({0..1/2::real} \<times> X)"
     apply (simp add: closedin_closed split_01_prod [symmetric])
     apply (rule_tac x="{0..1/2} \<times> UNIV" in exI)
     apply (force simp: closed_Times)
     done
-  have clo2: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({1/2..1::real} \<times> X)"
+  have clo2: "closedin (top_of_set ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({1/2..1::real} \<times> X)"
     apply (simp add: closedin_closed split_01_prod [symmetric])
     apply (rule_tac x="{1/2..1} \<times> UNIV" in exI)
     apply (force simp: closed_Times)
@@ -1472,20 +1472,20 @@
 definition%important locally :: "('a::topological_space set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
 where
  "locally P S \<equiv>
-        \<forall>w x. openin (subtopology euclidean S) w \<and> x \<in> w
-              \<longrightarrow> (\<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and>
+        \<forall>w x. openin (top_of_set S) w \<and> x \<in> w
+              \<longrightarrow> (\<exists>u v. openin (top_of_set S) u \<and> P v \<and>
                         x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)"
 
 lemma locallyI:
-  assumes "\<And>w x. \<lbrakk>openin (subtopology euclidean S) w; x \<in> w\<rbrakk>
-                  \<Longrightarrow> \<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and>
+  assumes "\<And>w x. \<lbrakk>openin (top_of_set S) w; x \<in> w\<rbrakk>
+                  \<Longrightarrow> \<exists>u v. openin (top_of_set S) u \<and> P v \<and>
                         x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w"
     shows "locally P S"
 using assms by (force simp: locally_def)
 
 lemma locallyE:
-  assumes "locally P S" "openin (subtopology euclidean S) w" "x \<in> w"
-  obtains u v where "openin (subtopology euclidean S) u"
+  assumes "locally P S" "openin (top_of_set S) w" "x \<in> w"
+  obtains u v where "openin (top_of_set S) u"
                     "P v" "x \<in> u" "u \<subseteq> v" "v \<subseteq> w"
   using assms unfolding locally_def by meson
 
@@ -1495,7 +1495,7 @@
 by (metis assms locally_def)
 
 lemma locally_open_subset:
-  assumes "locally P S" "openin (subtopology euclidean S) t"
+  assumes "locally P S" "openin (top_of_set S) t"
     shows "locally P t"
 using assms
 apply (simp add: locally_def)
@@ -1507,7 +1507,7 @@
 by (metis (no_types, hide_lams) Int_absorb1 Int_lower1 Int_subset_iff openin_open openin_subtopology_Int_subset)
 
 lemma locally_diff_closed:
-    "\<lbrakk>locally P S; closedin (subtopology euclidean S) t\<rbrakk> \<Longrightarrow> locally P (S - t)"
+    "\<lbrakk>locally P S; closedin (top_of_set S) t\<rbrakk> \<Longrightarrow> locally P (S - t)"
   using locally_open_subset closedin_def by fastforce
 
 lemma locally_empty [iff]: "locally P {}"
@@ -1550,15 +1550,15 @@
     unfolding locally_def
 proof (clarify)
   fix W x y
-  assume W: "openin (subtopology euclidean (S \<times> T)) W" and xy: "(x, y) \<in> W"
-  then obtain U V where "openin (subtopology euclidean S) U" "x \<in> U"
-                        "openin (subtopology euclidean T) V" "y \<in> V" "U \<times> V \<subseteq> W"
+  assume W: "openin (top_of_set (S \<times> T)) W" and xy: "(x, y) \<in> W"
+  then obtain U V where "openin (top_of_set S) U" "x \<in> U"
+                        "openin (top_of_set T) V" "y \<in> V" "U \<times> V \<subseteq> W"
     using Times_in_interior_subtopology by metis
   then obtain U1 U2 V1 V2
-         where opeS: "openin (subtopology euclidean S) U1 \<and> P U2 \<and> x \<in> U1 \<and> U1 \<subseteq> U2 \<and> U2 \<subseteq> U"
-           and opeT: "openin (subtopology euclidean T) V1 \<and> Q V2 \<and> y \<in> V1 \<and> V1 \<subseteq> V2 \<and> V2 \<subseteq> V"
+         where opeS: "openin (top_of_set S) U1 \<and> P U2 \<and> x \<in> U1 \<and> U1 \<subseteq> U2 \<and> U2 \<subseteq> U"
+           and opeT: "openin (top_of_set T) V1 \<and> Q V2 \<and> y \<in> V1 \<and> V1 \<subseteq> V2 \<and> V2 \<subseteq> V"
     by (meson PS QT locallyE)
-  with \<open>U \<times> V \<subseteq> W\<close> show "\<exists>u v. openin (subtopology euclidean (S \<times> T)) u \<and> R v \<and> (x,y) \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> W"
+  with \<open>U \<times> V \<subseteq> W\<close> show "\<exists>u v. openin (top_of_set (S \<times> T)) u \<and> R v \<and> (x,y) \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> W"
     apply (rule_tac x="U1 \<times> V1" in exI)
     apply (rule_tac x="U2 \<times> V2" in exI)
     apply (auto simp: openin_Times R)
@@ -1573,7 +1573,7 @@
     shows "locally Q t"
 proof (clarsimp simp: locally_def)
   fix W y
-  assume "y \<in> W" and "openin (subtopology euclidean t) W"
+  assume "y \<in> W" and "openin (top_of_set t) W"
   then obtain T where T: "open T" "W = t \<inter> T"
     by (force simp: openin_open)
   then have "W \<subseteq> t" by auto
@@ -1586,15 +1586,15 @@
     using \<open>g ` t = S\<close> \<open>W \<subseteq> t\<close> apply blast
     using g \<open>W \<subseteq> t\<close> apply auto[1]
     by (simp add: f rev_image_eqI)
-  have \<circ>: "openin (subtopology euclidean S) (g ` W)"
+  have \<circ>: "openin (top_of_set S) (g ` W)"
   proof -
     have "continuous_on S f"
       using f(3) by blast
-    then show "openin (subtopology euclidean S) (g ` W)"
-      by (simp add: gw Collect_conj_eq \<open>openin (subtopology euclidean t) W\<close> continuous_on_open f(2))
+    then show "openin (top_of_set S) (g ` W)"
+      by (simp add: gw Collect_conj_eq \<open>openin (top_of_set t) W\<close> continuous_on_open f(2))
   qed
   then obtain u v
-    where osu: "openin (subtopology euclidean S) u" and uv: "P v" "g y \<in> u" "u \<subseteq> v" "v \<subseteq> g ` W"
+    where osu: "openin (top_of_set S) u" and uv: "P v" "g y \<in> u" "u \<subseteq> v" "v \<subseteq> g ` W"
     using S [unfolded locally_def, rule_format, of "g ` W" "g y"] \<open>y \<in> W\<close> by force
   have "v \<subseteq> S" using uv by (simp add: gw)
   have fv: "f ` v = t \<inter> {x. g x \<in> v}"
@@ -1609,7 +1609,7 @@
     using \<open>v \<subseteq> S\<close> \<open>W \<subseteq> t\<close> f
     apply (simp add: homeomorphism_def contvf contvg, auto)
     by (metis f(1) rev_image_eqI rev_subsetD)
-  have 1: "openin (subtopology euclidean t) (t \<inter> g -` u)"
+  have 1: "openin (top_of_set t) (t \<inter> g -` u)"
     apply (rule continuous_on_open [THEN iffD1, rule_format])
     apply (rule \<open>continuous_on t g\<close>)
     using \<open>g ` t = S\<close> apply (simp add: osu)
@@ -1619,7 +1619,7 @@
     apply (intro conjI Q [OF \<open>P v\<close> homv])
     using \<open>W \<subseteq> t\<close> \<open>y \<in> W\<close>  \<open>f ` v \<subseteq> W\<close>  uv  apply (auto simp: fv)
     done
-  show "\<exists>U. openin (subtopology euclidean t) U \<and> (\<exists>v. Q v \<and> y \<in> U \<and> U \<subseteq> v \<and> v \<subseteq> W)"
+  show "\<exists>U. openin (top_of_set t) U \<and> (\<exists>v. Q v \<and> y \<in> U \<and> U \<subseteq> v \<and> v \<subseteq> W)"
     by (meson 1 2)
 qed
 
@@ -1674,23 +1674,23 @@
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes P: "locally P S"
       and f: "continuous_on S f"
-      and oo: "\<And>t. openin (subtopology euclidean S) t
-                   \<Longrightarrow> openin (subtopology euclidean (f ` S)) (f ` t)"
+      and oo: "\<And>t. openin (top_of_set S) t
+                   \<Longrightarrow> openin (top_of_set (f ` S)) (f ` t)"
       and Q: "\<And>t. \<lbrakk>t \<subseteq> S; P t\<rbrakk> \<Longrightarrow> Q(f ` t)"
     shows "locally Q (f ` S)"
 proof (clarsimp simp add: locally_def)
   fix W y
-  assume oiw: "openin (subtopology euclidean (f ` S)) W" and "y \<in> W"
+  assume oiw: "openin (top_of_set (f ` S)) W" and "y \<in> W"
   then have "W \<subseteq> f ` S" by (simp add: openin_euclidean_subtopology_iff)
-  have oivf: "openin (subtopology euclidean S) (S \<inter> f -` W)"
+  have oivf: "openin (top_of_set S) (S \<inter> f -` W)"
     by (rule continuous_on_open [THEN iffD1, rule_format, OF f oiw])
   then obtain x where "x \<in> S" "f x = y"
     using \<open>W \<subseteq> f ` S\<close> \<open>y \<in> W\<close> by blast
   then obtain U V
-    where "openin (subtopology euclidean S) U" "P V" "x \<in> U" "U \<subseteq> V" "V \<subseteq> S \<inter> f -` W"
+    where "openin (top_of_set S) U" "P V" "x \<in> U" "U \<subseteq> V" "V \<subseteq> S \<inter> f -` W"
     using P [unfolded locally_def, rule_format, of "(S \<inter> f -` W)" x] oivf \<open>y \<in> W\<close>
     by auto
-  then show "\<exists>X. openin (subtopology euclidean (f ` S)) X \<and> (\<exists>Y. Q Y \<and> y \<in> X \<and> X \<subseteq> Y \<and> Y \<subseteq> W)"
+  then show "\<exists>X. openin (top_of_set (f ` S)) X \<and> (\<exists>Y. Q Y \<and> y \<in> X \<and> X \<subseteq> Y \<and> Y \<subseteq> W)"
     apply (rule_tac x="f ` U" in exI)
     apply (rule conjI, blast intro!: oo)
     apply (rule_tac x="f ` V" in exI)
@@ -1703,21 +1703,21 @@
 
 proposition connected_induction:
   assumes "connected S"
-      and opD: "\<And>T a. \<lbrakk>openin (subtopology euclidean S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
+      and opD: "\<And>T a. \<lbrakk>openin (top_of_set S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
       and opI: "\<And>a. a \<in> S
-             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
+             \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and>
                      (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<and> Q x \<longrightarrow> Q y)"
       and etc: "a \<in> S" "b \<in> S" "P a" "P b" "Q a"
     shows "Q b"
 proof -
-  have 1: "openin (subtopology euclidean S)
-             {b. \<exists>T. openin (subtopology euclidean S) T \<and>
+  have 1: "openin (top_of_set S)
+             {b. \<exists>T. openin (top_of_set S) T \<and>
                      b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> Q x)}"
     apply (subst openin_subopen, clarify)
     apply (rule_tac x=T in exI, auto)
     done
-  have 2: "openin (subtopology euclidean S)
-             {b. \<exists>T. openin (subtopology euclidean S) T \<and>
+  have 2: "openin (top_of_set S)
+             {b. \<exists>T. openin (top_of_set S) T \<and>
                      b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> \<not> Q x)}"
     apply (subst openin_subopen, clarify)
     apply (rule_tac x=T in exI, auto)
@@ -1738,9 +1738,9 @@
   assumes "connected S"
       and etc: "a \<in> S" "b \<in> S" "P a" "P b"
       and trans: "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"
-      and opD: "\<And>T a. \<lbrakk>openin (subtopology euclidean S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
+      and opD: "\<And>T a. \<lbrakk>openin (top_of_set S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
       and opI: "\<And>a. a \<in> S
-             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
+             \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and>
                      (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<longrightarrow> R x y)"
     shows "R a b"
 proof -
@@ -1754,7 +1754,7 @@
   assumes "connected S"
       and etc: "a \<in> S" "b \<in> S" "P a"
       and opI: "\<And>a. a \<in> S
-             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
+             \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and>
                      (\<forall>x \<in> T. \<forall>y \<in> T. P x \<longrightarrow> P y)"
     shows "P b"
 apply (rule connected_induction [OF \<open>connected S\<close> _, where P = "\<lambda>x. True"], blast)
@@ -1767,7 +1767,7 @@
       and etc: "a \<in> S" "b \<in> S"
       and sym: "\<And>x y. \<lbrakk>R x y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> R y x"
       and trans: "\<And>x y z. \<lbrakk>R x y; R y z; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> R x z"
-      and opI: "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. R a x)"
+      and opI: "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. R a x)"
     shows "R a b"
 proof -
   have "\<And>a b c. \<lbrakk>a \<in> S; b \<in> S; c \<in> S; R a b\<rbrakk> \<Longrightarrow> R a c"
@@ -1779,7 +1779,7 @@
 lemma locally_constant_imp_constant:
   assumes "connected S"
       and opI: "\<And>a. a \<in> S
-             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. f x = f a)"
+             \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. f x = f a)"
     shows "f constant_on S"
 proof -
   have "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f x = f y"
@@ -1805,7 +1805,7 @@
   shows
     "locally compact s \<longleftrightarrow>
      (\<forall>x \<in> s. \<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
-                    openin (subtopology euclidean s) u \<and> compact v)"
+                    openin (top_of_set s) u \<and> compact v)"
      (is "?lhs = ?rhs")
 proof
   assume ?lhs
@@ -1816,11 +1816,11 @@
 next
   assume r [rule_format]: ?rhs
   have *: "\<exists>u v.
-              openin (subtopology euclidean s) u \<and>
+              openin (top_of_set s) u \<and>
               compact v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<inter> T"
           if "open T" "x \<in> s" "x \<in> T" for x T
   proof -
-    obtain u v where uv: "x \<in> u" "u \<subseteq> v" "v \<subseteq> s" "compact v" "openin (subtopology euclidean s) u"
+    obtain u v where uv: "x \<in> u" "u \<subseteq> v" "v \<subseteq> s" "compact v" "openin (top_of_set s) u"
       using r [OF \<open>x \<in> s\<close>] by auto
     obtain e where "e>0" and e: "cball x e \<subseteq> T"
       using open_contains_cball \<open>open T\<close> \<open>x \<in> T\<close> by blast
@@ -1842,7 +1842,7 @@
   fixes s :: "'a :: metric_space set"
   assumes "locally compact s"
   obtains u v where "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
-                             openin (subtopology euclidean s) (u x) \<and> compact (v x)"
+                             openin (top_of_set s) (u x) \<and> compact (v x)"
 using assms
 unfolding locally_compact by metis
 
@@ -1850,7 +1850,7 @@
   fixes s :: "'a :: heine_borel set"
   shows "locally compact s \<longleftrightarrow>
          (\<forall>x \<in> s. \<exists>u. x \<in> u \<and>
-                    openin (subtopology euclidean s) u \<and> compact(closure u) \<and> closure u \<subseteq> s)"
+                    openin (top_of_set s) u \<and> compact(closure u) \<and> closure u \<subseteq> s)"
 apply (simp add: locally_compact)
 apply (intro ball_cong ex_cong refl iffI)
 apply (metis bounded_subset closure_eq closure_mono compact_eq_bounded_closed dual_order.trans)
@@ -1884,21 +1884,21 @@
   shows "locally compact s \<longleftrightarrow>
          (\<forall>k. k \<subseteq> s \<and> compact k
               \<longrightarrow> (\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
-                         openin (subtopology euclidean s) u \<and> compact v))"
+                         openin (top_of_set s) u \<and> compact v))"
         (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then obtain u v where
     uv: "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
-                             openin (subtopology euclidean s) (u x) \<and> compact (v x)"
+                             openin (top_of_set s) (u x) \<and> compact (v x)"
     by (metis locally_compactE)
-  have *: "\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v"
+  have *: "\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (top_of_set s) u \<and> compact v"
           if "k \<subseteq> s" "compact k" for k
   proof -
-    have "\<And>C. (\<forall>c\<in>C. openin (subtopology euclidean k) c) \<and> k \<subseteq> \<Union>C \<Longrightarrow>
+    have "\<And>C. (\<forall>c\<in>C. openin (top_of_set k) c) \<and> k \<subseteq> \<Union>C \<Longrightarrow>
                     \<exists>D\<subseteq>C. finite D \<and> k \<subseteq> \<Union>D"
       using that by (simp add: compact_eq_openin_cover)
-    moreover have "\<forall>c \<in> (\<lambda>x. k \<inter> u x) ` k. openin (subtopology euclidean k) c"
+    moreover have "\<forall>c \<in> (\<lambda>x. k \<inter> u x) ` k. openin (top_of_set k) c"
       using that by clarify (metis subsetD inf.absorb_iff2 openin_subset openin_subtopology_Int_subset topspace_euclidean_subtopology uv)
     moreover have "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` k)"
       using that by clarsimp (meson subsetCE uv)
@@ -1931,12 +1931,12 @@
   assumes "open s"
     shows "locally compact s"
 proof -
-  have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v"
+  have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (top_of_set s) u \<and> compact v"
           if "x \<in> s" for x
   proof -
     obtain e where "e>0" and e: "cball x e \<subseteq> s"
       using open_contains_cball assms \<open>x \<in> s\<close> by blast
-    have ope: "openin (subtopology euclidean s) (ball x e)"
+    have ope: "openin (top_of_set s) (ball x e)"
       by (meson e open_ball ball_subset_cball dual_order.trans open_subset)
     show ?thesis
       apply (rule_tac x="ball x e" in exI)
@@ -1955,7 +1955,7 @@
     shows "locally compact s"
 proof -
   have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
-                 openin (subtopology euclidean s) u \<and> compact v"
+                 openin (top_of_set s) u \<and> compact v"
           if "x \<in> s" for x
   proof -
     show ?thesis
@@ -1979,7 +1979,7 @@
 
 lemma locally_compact_closedin:
   fixes s :: "'a :: heine_borel set"
-  shows "\<lbrakk>closedin (subtopology euclidean s) t; locally compact s\<rbrakk>
+  shows "\<lbrakk>closedin (top_of_set s) t; locally compact s\<rbrakk>
         \<Longrightarrow> locally compact t"
 unfolding closedin_closed
 using closed_imp_locally_compact locally_compact_Int by blast
@@ -2010,8 +2010,8 @@
 lemma locally_compact_openin_Un:
   fixes S :: "'a::euclidean_space set"
   assumes LCS: "locally compact S" and LCT:"locally compact T"
-      and opS: "openin (subtopology euclidean (S \<union> T)) S"
-      and opT: "openin (subtopology euclidean (S \<union> T)) T"
+      and opS: "openin (top_of_set (S \<union> T)) S"
+      and opT: "openin (top_of_set (S \<union> T)) T"
     shows "locally compact (S \<union> T)"
 proof -
   have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" for x
@@ -2047,8 +2047,8 @@
 lemma locally_compact_closedin_Un:
   fixes S :: "'a::euclidean_space set"
   assumes LCS: "locally compact S" and LCT:"locally compact T"
-      and clS: "closedin (subtopology euclidean (S \<union> T)) S"
-      and clT: "closedin (subtopology euclidean (S \<union> T)) T"
+      and clS: "closedin (top_of_set (S \<union> T)) S"
+      and clT: "closedin (top_of_set (S \<union> T)) T"
     shows "locally compact (S \<union> T)"
 proof -
   have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" "x \<in> T" for x
@@ -2113,7 +2113,7 @@
    "locally compact S \<longleftrightarrow>
     (\<forall>K T. K \<subseteq> S \<and> compact K \<and> open T \<and> K \<subseteq> T
           \<longrightarrow> (\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
-                     openin (subtopology euclidean S) U \<and> compact V))"
+                     openin (top_of_set S) U \<and> compact V))"
    (is "?lhs = ?rhs")
 proof
   assume L: ?lhs
@@ -2122,10 +2122,10 @@
     fix K :: "'a set" and T :: "'a set"
     assume "K \<subseteq> S" and "compact K" and "open T" and "K \<subseteq> T"
     obtain U V where "K \<subseteq> U" "U \<subseteq> V" "V \<subseteq> S" "compact V"
-                 and ope: "openin (subtopology euclidean S) U"
+                 and ope: "openin (top_of_set S) U"
       using L unfolding locally_compact_compact by (meson \<open>K \<subseteq> S\<close> \<open>compact K\<close>)
     show "\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
-                openin (subtopology euclidean S) U \<and> compact V"
+                openin (top_of_set S) U \<and> compact V"
     proof (intro exI conjI)
       show "K \<subseteq> U \<inter> T"
         by (simp add: \<open>K \<subseteq> T\<close> \<open>K \<subseteq> U\<close>)
@@ -2133,7 +2133,7 @@
         by (rule closure_subset)
       show "closure (U \<inter> T) \<subseteq> S"
         by (metis \<open>U \<subseteq> V\<close> \<open>V \<subseteq> S\<close> \<open>compact V\<close> closure_closed closure_mono compact_imp_closed inf.cobounded1 subset_trans)
-      show "openin (subtopology euclidean S) (U \<inter> T)"
+      show "openin (top_of_set S) (U \<inter> T)"
         by (simp add: \<open>open T\<close> ope openin_Int_open)
       show "compact (closure (U \<inter> T))"
         by (meson Int_lower1 \<open>U \<subseteq> V\<close> \<open>compact V\<close> bounded_subset compact_closure compact_eq_bounded_closed)
@@ -2151,8 +2151,8 @@
 proposition Sura_Bura_compact:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" and C: "C \<in> components S"
-  shows "C = \<Inter>{T. C \<subseteq> T \<and> openin (subtopology euclidean S) T \<and>
-                           closedin (subtopology euclidean S) T}"
+  shows "C = \<Inter>{T. C \<subseteq> T \<and> openin (top_of_set S) T \<and>
+                           closedin (top_of_set S) T}"
          (is "C = \<Inter>?\<T>")
 proof
   obtain x where x: "C = connected_component_set S x" and "x \<in> S"
@@ -2168,8 +2168,8 @@
     have clo: "closed (\<Inter>?\<T>)"
       by (simp add: \<open>compact S\<close> closed_Inter closedin_compact_eq compact_imp_closed)
     have False
-      if K1: "closedin (subtopology euclidean (\<Inter>?\<T>)) K1" and
-         K2: "closedin (subtopology euclidean (\<Inter>?\<T>)) K2" and
+      if K1: "closedin (top_of_set (\<Inter>?\<T>)) K1" and
+         K2: "closedin (top_of_set (\<Inter>?\<T>)) K2" and
          K12_Int: "K1 \<inter> K2 = {}" and K12_Un: "K1 \<union> K2 = \<Inter>?\<T>" and "K1 \<noteq> {}" "K2 \<noteq> {}"
        for K1 K2
     proof -
@@ -2187,8 +2187,8 @@
         show "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> \<noteq> {}" if "finite \<F>" and \<F>: "\<F> \<subseteq> ?\<T>" for \<F>
         proof
           assume djo: "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> = {}"
-          obtain D where opeD: "openin (subtopology euclidean S) D"
-                   and cloD: "closedin (subtopology euclidean S) D"
+          obtain D where opeD: "openin (top_of_set S) D"
+                   and cloD: "closedin (top_of_set S) D"
                    and "C \<subseteq> D" and DV12: "D \<subseteq> V1 \<union> V2"
           proof (cases "\<F> = {}")
             case True
@@ -2197,9 +2197,9 @@
           next
             case False show ?thesis
             proof
-              show ope: "openin (subtopology euclidean S) (\<Inter>\<F>)"
+              show ope: "openin (top_of_set S) (\<Inter>\<F>)"
                 using openin_Inter \<open>finite \<F>\<close> False \<F> by blast
-              then show "closedin (subtopology euclidean S) (\<Inter>\<F>)"
+              then show "closedin (top_of_set S) (\<Inter>\<F>)"
                 by (meson clo\<T> \<F> closed_Inter closed_subset openin_imp_subset subset_eq)
               show "C \<subseteq> \<Inter>\<F>"
                 using \<F> by auto
@@ -2211,16 +2211,16 @@
             by (simp add: x)
           have "closed D"
             using \<open>compact S\<close> cloD closedin_closed_trans compact_imp_closed by blast
-          have cloV1: "closedin (subtopology euclidean D) (D \<inter> closure V1)"
-            and cloV2: "closedin (subtopology euclidean D) (D \<inter> closure V2)"
+          have cloV1: "closedin (top_of_set D) (D \<inter> closure V1)"
+            and cloV2: "closedin (top_of_set D) (D \<inter> closure V2)"
             by (simp_all add: closedin_closed_Int)
           moreover have "D \<inter> closure V1 = D \<inter> V1" "D \<inter> closure V2 = D \<inter> V2"
             apply safe
             using \<open>D \<subseteq> V1 \<union> V2\<close> \<open>open V1\<close> \<open>open V2\<close> V12
                apply (simp_all add: closure_subset [THEN subsetD] closure_iff_nhds_not_empty, blast+)
             done
-          ultimately have cloDV1: "closedin (subtopology euclidean D) (D \<inter> V1)"
-                      and cloDV2:  "closedin (subtopology euclidean D) (D \<inter> V2)"
+          ultimately have cloDV1: "closedin (top_of_set D) (D \<inter> V1)"
+                      and cloDV2:  "closedin (top_of_set D) (D \<inter> V2)"
             by metis+
           then obtain U1 U2 where "closed U1" "closed U2"
                and D1: "D \<inter> V1 = D \<inter> U1" and D2: "D \<inter> V2 = D \<inter> U2"
@@ -2274,21 +2274,21 @@
   fixes S :: "'a::euclidean_space set"
   assumes S: "locally compact S" and C: "C \<in> components S" and "compact C"
       and U: "open U" "C \<subseteq> U"
-  obtains K where "openin (subtopology euclidean S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
+  obtains K where "openin (top_of_set S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
 proof (rule ccontr)
   assume "\<not> thesis"
-  with that have neg: "\<nexists>K. openin (subtopology euclidean S) K \<and> compact K \<and> C \<subseteq> K \<and> K \<subseteq> U"
+  with that have neg: "\<nexists>K. openin (top_of_set S) K \<and> compact K \<and> C \<subseteq> K \<and> K \<subseteq> U"
     by metis
   obtain V K where "C \<subseteq> V" "V \<subseteq> U" "V \<subseteq> K" "K \<subseteq> S" "compact K"
-               and opeSV: "openin (subtopology euclidean S) V"
+               and opeSV: "openin (top_of_set S) V"
     using S U \<open>compact C\<close>
     apply (simp add: locally_compact_compact_subopen)
     by (meson C in_components_subset)
-  let ?\<T> = "{T. C \<subseteq> T \<and> openin (subtopology euclidean K) T \<and> compact T \<and> T \<subseteq> K}"
+  let ?\<T> = "{T. C \<subseteq> T \<and> openin (top_of_set K) T \<and> compact T \<and> T \<subseteq> K}"
   have CK: "C \<in> components K"
     by (meson C \<open>C \<subseteq> V\<close> \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> components_intermediate_subset subset_trans)
   with \<open>compact K\<close>
-  have "C = \<Inter>{T. C \<subseteq> T \<and> openin (subtopology euclidean K) T \<and> closedin (subtopology euclidean K) T}"
+  have "C = \<Inter>{T. C \<subseteq> T \<and> openin (top_of_set K) T \<and> closedin (top_of_set K) T}"
     by (simp add: Sura_Bura_compact)
   then have Ceq: "C = \<Inter>?\<T>"
     by (simp add: closedin_compact_eq \<open>compact K\<close>)
@@ -2322,11 +2322,11 @@
           by (metis (no_types, lifting) compact_Inter False mem_Collect_eq subsetCE \<F>)
         moreover have "\<Inter>\<F> \<subseteq> K"
           using False that(2) by fastforce
-        moreover have opeKF: "openin (subtopology euclidean K) (\<Inter>\<F>)"
+        moreover have opeKF: "openin (top_of_set K) (\<Inter>\<F>)"
           using False \<F> \<open>finite \<F>\<close> by blast
-        then have opeVF: "openin (subtopology euclidean V) (\<Inter>\<F>)"
+        then have opeVF: "openin (top_of_set V) (\<Inter>\<F>)"
           using W \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> opeKF \<open>\<Inter>\<F> \<subseteq> K\<close> FUW openin_subset_trans by fastforce
-        then have "openin (subtopology euclidean S) (\<Inter>\<F>)"
+        then have "openin (top_of_set S) (\<Inter>\<F>)"
           by (metis opeSV openin_trans)
         moreover have "\<Inter>\<F> \<subseteq> U"
           by (meson \<open>V \<subseteq> U\<close> opeVF dual_order.trans openin_imp_subset)
@@ -2343,8 +2343,8 @@
 corollary Sura_Bura_clopen_subset_alt:
   fixes S :: "'a::euclidean_space set"
   assumes S: "locally compact S" and C: "C \<in> components S" and "compact C"
-      and opeSU: "openin (subtopology euclidean S) U" and "C \<subseteq> U"
-  obtains K where "openin (subtopology euclidean S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
+      and opeSU: "openin (top_of_set S) U" and "C \<subseteq> U"
+  obtains K where "openin (top_of_set S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
 proof -
   obtain V where "open V" "U = S \<inter> V"
     using opeSU by (auto simp: openin_open)
@@ -2358,13 +2358,13 @@
 corollary Sura_Bura:
   fixes S :: "'a::euclidean_space set"
   assumes "locally compact S" "C \<in> components S" "compact C"
-  shows "C = \<Inter> {K. C \<subseteq> K \<and> compact K \<and> openin (subtopology euclidean S) K}"
+  shows "C = \<Inter> {K. C \<subseteq> K \<and> compact K \<and> openin (top_of_set S) K}"
          (is "C = ?rhs")
 proof
   show "?rhs \<subseteq> C"
   proof (clarsimp, rule ccontr)
     fix x
-    assume *: "\<forall>X. C \<subseteq> X \<and> compact X \<and> openin (subtopology euclidean S) X \<longrightarrow> x \<in> X"
+    assume *: "\<forall>X. C \<subseteq> X \<and> compact X \<and> openin (top_of_set S) X \<longrightarrow> x \<in> X"
       and "x \<notin> C"
     obtain U V where "open U" "open V" "{x} \<subseteq> U" "C \<subseteq> V" "U \<inter> V = {}"
       using separation_normal [of "{x}" C]
@@ -2381,8 +2381,8 @@
 
 lemma locally_connected_1:
   assumes
-    "\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk>
-              \<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and>
+    "\<And>v x. \<lbrakk>openin (top_of_set S) v; x \<in> v\<rbrakk>
+              \<Longrightarrow> \<exists>u. openin (top_of_set S) u \<and>
                       connected u \<and> x \<in> u \<and> u \<subseteq> v"
    shows "locally connected S"
 apply (clarsimp simp add: locally_def)
@@ -2391,12 +2391,12 @@
 
 lemma locally_connected_2:
   assumes "locally connected S"
-          "openin (subtopology euclidean S) t"
+          "openin (top_of_set S) t"
           "x \<in> t"
-   shows "openin (subtopology euclidean S) (connected_component_set t x)"
+   shows "openin (top_of_set S) (connected_component_set t x)"
 proof -
   { fix y :: 'a
-    let ?SS = "subtopology euclidean S"
+    let ?SS = "top_of_set S"
     assume 1: "openin ?SS t"
               "\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))"
     and "connected_component t x y"
@@ -2420,29 +2420,29 @@
 qed
 
 lemma locally_connected_3:
-  assumes "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk>
-              \<Longrightarrow> openin (subtopology euclidean S)
+  assumes "\<And>t x. \<lbrakk>openin (top_of_set S) t; x \<in> t\<rbrakk>
+              \<Longrightarrow> openin (top_of_set S)
                           (connected_component_set t x)"
-          "openin (subtopology euclidean S) v" "x \<in> v"
-   shows  "\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v"
+          "openin (top_of_set S) v" "x \<in> v"
+   shows  "\<exists>u. openin (top_of_set S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v"
 using assms connected_component_subset by fastforce
 
 lemma locally_connected:
   "locally connected S \<longleftrightarrow>
-   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
-          \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v))"
+   (\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
+          \<longrightarrow> (\<exists>u. openin (top_of_set S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v))"
 by (metis locally_connected_1 locally_connected_2 locally_connected_3)
 
 lemma locally_connected_open_connected_component:
   "locally connected S \<longleftrightarrow>
-   (\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t
-          \<longrightarrow> openin (subtopology euclidean S) (connected_component_set t x))"
+   (\<forall>t x. openin (top_of_set S) t \<and> x \<in> t
+          \<longrightarrow> openin (top_of_set S) (connected_component_set t x))"
 by (metis locally_connected_1 locally_connected_2 locally_connected_3)
 
 lemma locally_path_connected_1:
   assumes
-    "\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk>
-              \<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
+    "\<And>v x. \<lbrakk>openin (top_of_set S) v; x \<in> v\<rbrakk>
+              \<Longrightarrow> \<exists>u. openin (top_of_set S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
    shows "locally path_connected S"
 apply (clarsimp simp add: locally_def)
 apply (drule assms; blast)
@@ -2450,12 +2450,12 @@
 
 lemma locally_path_connected_2:
   assumes "locally path_connected S"
-          "openin (subtopology euclidean S) t"
+          "openin (top_of_set S) t"
           "x \<in> t"
-   shows "openin (subtopology euclidean S) (path_component_set t x)"
+   shows "openin (top_of_set S) (path_component_set t x)"
 proof -
   { fix y :: 'a
-    let ?SS = "subtopology euclidean S"
+    let ?SS = "top_of_set S"
     assume 1: "openin ?SS t"
               "\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. path_connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))"
     and "path_component t x y"
@@ -2479,10 +2479,10 @@
 qed
 
 lemma locally_path_connected_3:
-  assumes "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk>
-              \<Longrightarrow> openin (subtopology euclidean S) (path_component_set t x)"
-          "openin (subtopology euclidean S) v" "x \<in> v"
-   shows  "\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
+  assumes "\<And>t x. \<lbrakk>openin (top_of_set S) t; x \<in> t\<rbrakk>
+              \<Longrightarrow> openin (top_of_set S) (path_component_set t x)"
+          "openin (top_of_set S) v" "x \<in> v"
+   shows  "\<exists>u. openin (top_of_set S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
 proof -
   have "path_component v x x"
     by (meson assms(3) path_component_refl)
@@ -2492,26 +2492,26 @@
 
 proposition locally_path_connected:
   "locally path_connected S \<longleftrightarrow>
-   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
-          \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v))"
+   (\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
+          \<longrightarrow> (\<exists>u. openin (top_of_set S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v))"
   by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)
 
 proposition locally_path_connected_open_path_component:
   "locally path_connected S \<longleftrightarrow>
-   (\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t
-          \<longrightarrow> openin (subtopology euclidean S) (path_component_set t x))"
+   (\<forall>t x. openin (top_of_set S) t \<and> x \<in> t
+          \<longrightarrow> openin (top_of_set S) (path_component_set t x))"
   by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)
 
 lemma locally_connected_open_component:
   "locally connected S \<longleftrightarrow>
-   (\<forall>t c. openin (subtopology euclidean S) t \<and> c \<in> components t
-          \<longrightarrow> openin (subtopology euclidean S) c)"
+   (\<forall>t c. openin (top_of_set S) t \<and> c \<in> components t
+          \<longrightarrow> openin (top_of_set S) c)"
 by (metis components_iff locally_connected_open_connected_component)
 
 proposition locally_connected_im_kleinen:
   "locally connected S \<longleftrightarrow>
-   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
-       \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and>
+   (\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
+       \<longrightarrow> (\<exists>u. openin (top_of_set S) u \<and>
                 x \<in> u \<and> u \<subseteq> v \<and>
                 (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> v \<and> x \<in> c \<and> y \<in> c))))"
    (is "?lhs = ?rhs")
@@ -2521,12 +2521,12 @@
     by (fastforce simp add: locally_connected)
 next
   assume ?rhs
-  have *: "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> c"
-       if "openin (subtopology euclidean S) t" and c: "c \<in> components t" and "x \<in> c" for t c x
+  have *: "\<exists>T. openin (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> c"
+       if "openin (top_of_set S) t" and c: "c \<in> components t" and "x \<in> c" for t c x
   proof -
     from that \<open>?rhs\<close> [rule_format, of t x]
     obtain u where u:
-      "openin (subtopology euclidean S) u \<and> x \<in> u \<and> u \<subseteq> t \<and>
+      "openin (top_of_set S) u \<and> x \<in> u \<and> u \<subseteq> t \<and>
        (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> t \<and> x \<in> c \<and> y \<in> c))"
       using in_components_subset by auto
     obtain F :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where
@@ -2551,8 +2551,8 @@
 
 proposition locally_path_connected_im_kleinen:
   "locally path_connected S \<longleftrightarrow>
-   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
-       \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and>
+   (\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
+       \<longrightarrow> (\<exists>u. openin (top_of_set S) u \<and>
                 x \<in> u \<and> u \<subseteq> v \<and>
                 (\<forall>y. y \<in> u \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> v \<and>
                                 pathstart p = x \<and> pathfinish p = y))))"
@@ -2565,15 +2565,15 @@
     by (meson dual_order.trans)
 next
   assume ?rhs
-  have *: "\<exists>T. openin (subtopology euclidean S) T \<and>
+  have *: "\<exists>T. openin (top_of_set S) T \<and>
                x \<in> T \<and> T \<subseteq> path_component_set u z"
-       if "openin (subtopology euclidean S) u" and "z \<in> u" and c: "path_component u z x" for u z x
+       if "openin (top_of_set S) u" and "z \<in> u" and c: "path_component u z x" for u z x
   proof -
     have "x \<in> u"
       by (meson c path_component_mem(2))
     with that \<open>?rhs\<close> [rule_format, of u x]
     obtain U where U:
-      "openin (subtopology euclidean S) U \<and> x \<in> U \<and> U \<subseteq> u \<and>
+      "openin (top_of_set S) U \<and> x \<in> U \<and> U \<subseteq> u \<and>
        (\<forall>y. y \<in> U \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> u \<and> pathstart p = x \<and> pathfinish p = y))"
        by blast
     show ?thesis
@@ -2626,22 +2626,22 @@
 
 lemma openin_connected_component_locally_connected:
     "locally connected S
-     \<Longrightarrow> openin (subtopology euclidean S) (connected_component_set S x)"
+     \<Longrightarrow> openin (top_of_set S) (connected_component_set S x)"
 apply (simp add: locally_connected_open_connected_component)
 by (metis connected_component_eq_empty connected_component_subset open_empty open_subset openin_subtopology_self)
 
 lemma openin_components_locally_connected:
-    "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> openin (subtopology euclidean S) c"
+    "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> openin (top_of_set S) c"
   using locally_connected_open_component openin_subtopology_self by blast
 
 lemma openin_path_component_locally_path_connected:
   "locally path_connected S
-        \<Longrightarrow> openin (subtopology euclidean S) (path_component_set S x)"
+        \<Longrightarrow> openin (top_of_set S) (path_component_set S x)"
 by (metis (no_types) empty_iff locally_path_connected_2 openin_subopen openin_subtopology_self path_component_eq_empty)
 
 lemma closedin_path_component_locally_path_connected:
     "locally path_connected S
-        \<Longrightarrow> closedin (subtopology euclidean S) (path_component_set S x)"
+        \<Longrightarrow> closedin (top_of_set S) (path_component_set S x)"
 apply  (simp add: closedin_def path_component_subset complement_path_component_Union)
 apply (rule openin_Union)
 using openin_path_component_locally_path_connected by auto
@@ -2670,12 +2670,12 @@
     shows "(path_component S x = connected_component S x)"
 proof (cases "x \<in> S")
   case True
-  have "openin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
+  have "openin (top_of_set (connected_component_set S x)) (path_component_set S x)"
     apply (rule openin_subset_trans [of S])
     apply (intro conjI openin_path_component_locally_path_connected [OF assms])
     using path_component_subset_connected_component   apply (auto simp: connected_component_subset)
     done
-  moreover have "closedin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
+  moreover have "closedin (top_of_set (connected_component_set S x)) (path_component_set S x)"
     apply (rule closedin_subset_trans [of S])
     apply (intro conjI closedin_path_component_locally_path_connected [OF assms])
     using path_component_subset_connected_component   apply (auto simp: connected_component_subset)
@@ -2710,26 +2710,26 @@
 proposition locally_connected_quotient_image:
   assumes lcS: "locally connected S"
       and oo: "\<And>T. T \<subseteq> f ` S
-                \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow>
-                    openin (subtopology euclidean (f ` S)) T"
+                \<Longrightarrow> openin (top_of_set S) (S \<inter> f -` T) \<longleftrightarrow>
+                    openin (top_of_set (f ` S)) T"
     shows "locally connected (f ` S)"
 proof (clarsimp simp: locally_connected_open_component)
   fix U C
-  assume opefSU: "openin (subtopology euclidean (f ` S)) U" and "C \<in> components U"
+  assume opefSU: "openin (top_of_set (f ` S)) U" and "C \<in> components U"
   then have "C \<subseteq> U" "U \<subseteq> f ` S"
     by (meson in_components_subset openin_imp_subset)+
-  then have "openin (subtopology euclidean (f ` S)) C \<longleftrightarrow>
-             openin (subtopology euclidean S) (S \<inter> f -` C)"
+  then have "openin (top_of_set (f ` S)) C \<longleftrightarrow>
+             openin (top_of_set S) (S \<inter> f -` C)"
     by (auto simp: oo)
-  moreover have "openin (subtopology euclidean S) (S \<inter> f -` C)"
+  moreover have "openin (top_of_set S) (S \<inter> f -` C)"
   proof (subst openin_subopen, clarify)
     fix x
     assume "x \<in> S" "f x \<in> C"
-    show "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` C)"
+    show "\<exists>T. openin (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` C)"
     proof (intro conjI exI)
-      show "openin (subtopology euclidean S) (connected_component_set (S \<inter> f -` U) x)"
+      show "openin (top_of_set S) (connected_component_set (S \<inter> f -` U) x)"
       proof (rule ccontr)
-        assume **: "\<not> openin (subtopology euclidean S) (connected_component_set (S \<inter> f -` U) x)"
+        assume **: "\<not> openin (top_of_set S) (connected_component_set (S \<inter> f -` U) x)"
         then have "x \<notin> (S \<inter> f -` U)"
           using \<open>U \<subseteq> f ` S\<close> opefSU lcS locally_connected_2 oo by blast
         with ** show False
@@ -2768,7 +2768,7 @@
       finally show "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)" .
     qed
   qed
-  ultimately show "openin (subtopology euclidean (f ` S)) C"
+  ultimately show "openin (top_of_set (f ` S)) C"
     by metis
 qed
 
@@ -2776,32 +2776,32 @@
 proposition locally_path_connected_quotient_image:
   assumes lcS: "locally path_connected S"
       and oo: "\<And>T. T \<subseteq> f ` S
-                \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow> openin (subtopology euclidean (f ` S)) T"
+                \<Longrightarrow> openin (top_of_set S) (S \<inter> f -` T) \<longleftrightarrow> openin (top_of_set (f ` S)) T"
     shows "locally path_connected (f ` S)"
 proof (clarsimp simp: locally_path_connected_open_path_component)
   fix U y
-  assume opefSU: "openin (subtopology euclidean (f ` S)) U" and "y \<in> U"
+  assume opefSU: "openin (top_of_set (f ` S)) U" and "y \<in> U"
   then have "path_component_set U y \<subseteq> U" "U \<subseteq> f ` S"
     by (meson path_component_subset openin_imp_subset)+
-  then have "openin (subtopology euclidean (f ` S)) (path_component_set U y) \<longleftrightarrow>
-             openin (subtopology euclidean S) (S \<inter> f -` path_component_set U y)"
+  then have "openin (top_of_set (f ` S)) (path_component_set U y) \<longleftrightarrow>
+             openin (top_of_set S) (S \<inter> f -` path_component_set U y)"
   proof -
     have "path_component_set U y \<subseteq> f ` S"
       using \<open>U \<subseteq> f ` S\<close> \<open>path_component_set U y \<subseteq> U\<close> by blast
     then show ?thesis
       using oo by blast
   qed
-  moreover have "openin (subtopology euclidean S) (S \<inter> f -` path_component_set U y)"
+  moreover have "openin (top_of_set S) (S \<inter> f -` path_component_set U y)"
   proof (subst openin_subopen, clarify)
     fix x
     assume "x \<in> S" and Uyfx: "path_component U y (f x)"
     then have "f x \<in> U"
       using path_component_mem by blast
-    show "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` path_component_set U y)"
+    show "\<exists>T. openin (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` path_component_set U y)"
     proof (intro conjI exI)
-      show "openin (subtopology euclidean S) (path_component_set (S \<inter> f -` U) x)"
+      show "openin (top_of_set S) (path_component_set (S \<inter> f -` U) x)"
       proof (rule ccontr)
-        assume **: "\<not> openin (subtopology euclidean S) (path_component_set (S \<inter> f -` U) x)"
+        assume **: "\<not> openin (top_of_set S) (path_component_set (S \<inter> f -` U) x)"
         then have "x \<notin> (S \<inter> f -` U)"
           by (metis (no_types, lifting) \<open>U \<subseteq> f ` S\<close> opefSU lcS oo locally_path_connected_open_path_component)
         then show False
@@ -2842,7 +2842,7 @@
       finally show "path_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` path_component_set U y)" .
     qed
   qed
-  ultimately show "openin (subtopology euclidean (f ` S)) (path_component_set U y)"
+  ultimately show "openin (top_of_set (f ` S)) (path_component_set U y)"
     by metis
 qed
 
@@ -2851,14 +2851,14 @@
 lemma continuous_on_components_gen:
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
   assumes "\<And>c. c \<in> components S \<Longrightarrow>
-              openin (subtopology euclidean S) c \<and> continuous_on c f"
+              openin (top_of_set S) c \<and> continuous_on c f"
     shows "continuous_on S f"
 proof (clarsimp simp: continuous_openin_preimage_eq)
   fix t :: "'b set"
   assume "open t"
   have *: "S \<inter> f -` t = (\<Union>c \<in> components S. c \<inter> f -` t)"
     by auto
-  show "openin (subtopology euclidean S) (S \<inter> f -` t)"
+  show "openin (top_of_set S) (S \<inter> f -` t)"
     unfolding * using \<open>open t\<close> assms continuous_openin_preimage_gen openin_trans openin_Union by blast
 qed
 
@@ -2891,9 +2891,9 @@
 
 lemma closedin_union_complement_components:
   assumes u: "locally connected u"
-      and S: "closedin (subtopology euclidean u) S"
+      and S: "closedin (top_of_set u) S"
       and cuS: "c \<subseteq> components(u - S)"
-    shows "closedin (subtopology euclidean u) (S \<union> \<Union>c)"
+    shows "closedin (top_of_set u) (S \<union> \<Union>c)"
 proof -
   have di: "(\<And>S t. S \<in> c \<and> t \<in> c' \<Longrightarrow> disjnt S t) \<Longrightarrow> disjnt (\<Union> c) (\<Union> c')" for c'
     by (simp add: disjnt_def) blast
@@ -2906,13 +2906,13 @@
     by (metis DiffD1 DiffD2 assms(3) components_nonoverlap disjnt_def subsetCE)
   ultimately have eq: "S \<union> \<Union>c = u - (\<Union>(components(u - S) - c))"
     by (auto simp: disjnt_def)
-  have *: "openin (subtopology euclidean u) (\<Union>(components (u - S) - c))"
+  have *: "openin (top_of_set u) (\<Union>(components (u - S) - c))"
     apply (rule openin_Union)
     apply (rule openin_trans [of "u - S"])
     apply (simp add: u S locally_diff_closed openin_components_locally_connected)
     apply (simp add: openin_diff S)
     done
-  have "openin (subtopology euclidean u) (u - (u - \<Union>(components (u - S) - c)))"
+  have "openin (top_of_set u) (u - (u - \<Union>(components (u - S) - c)))"
     apply (rule openin_diff, simp)
     apply (metis closedin_diff closedin_topspace topspace_euclidean_subtopology *)
     done
@@ -2925,7 +2925,7 @@
   assumes S: "closed S" and c: "c \<subseteq> components(- S)"
     shows "closed(S \<union> \<Union> c)"
 proof -
-  have "closedin (subtopology euclidean UNIV) (S \<union> \<Union>c)"
+  have "closedin (top_of_set UNIV) (S \<union> \<Union>c)"
     apply (rule closedin_union_complement_components [OF locally_connected_UNIV])
     using S c apply (simp_all add: Compl_eq_Diff_UNIV)
     done
@@ -2935,11 +2935,11 @@
 lemma closedin_Un_complement_component:
   fixes S :: "'a::real_normed_vector set"
   assumes u: "locally connected u"
-      and S: "closedin (subtopology euclidean u) S"
+      and S: "closedin (top_of_set u) S"
       and c: " c \<in> components(u - S)"
-    shows "closedin (subtopology euclidean u) (S \<union> c)"
+    shows "closedin (top_of_set u) (S \<union> c)"
 proof -
-  have "closedin (subtopology euclidean u) (S \<union> \<Union>{c})"
+  have "closedin (top_of_set u) (S \<union> \<Union>{c})"
     using c by (blast intro: closedin_union_complement_components [OF u S])
   then show ?thesis
     by simp
@@ -3987,7 +3987,7 @@
 
 proposition path_connected_openin_diff_countable:
   fixes S :: "'a::euclidean_space set"
-  assumes "connected S" and ope: "openin (subtopology euclidean (affine hull S)) S"
+  assumes "connected S" and ope: "openin (top_of_set (affine hull S)) S"
       and "\<not> collinear S" "countable T"
     shows "path_connected(S - T)"
 proof (clarsimp simp add: path_connected_component)
@@ -3995,9 +3995,9 @@
   assume xy: "x \<in> S" "x \<notin> T" "y \<in> S" "y \<notin> T"
   show "path_component (S - T) x y"
   proof (rule connected_equivalence_relation_gen [OF \<open>connected S\<close>, where P = "\<lambda>x. x \<notin> T"])
-    show "\<exists>z. z \<in> U \<and> z \<notin> T" if opeU: "openin (subtopology euclidean S) U" and "x \<in> U" for U x
+    show "\<exists>z. z \<in> U \<and> z \<notin> T" if opeU: "openin (top_of_set S) U" and "x \<in> U" for U x
     proof -
-      have "openin (subtopology euclidean (affine hull S)) U"
+      have "openin (top_of_set (affine hull S)) U"
         using opeU ope openin_trans by blast
       with \<open>x \<in> U\<close> obtain r where Usub: "U \<subseteq> affine hull S" and "r > 0"
                               and subU: "ball x r \<inter> affine hull S \<subseteq> U"
@@ -4024,7 +4024,7 @@
         using \<open>countable T\<close> countable_subset by blast
       then show ?thesis by blast
     qed
-    show "\<exists>U. openin (subtopology euclidean S) U \<and> x \<in> U \<and>
+    show "\<exists>U. openin (top_of_set S) U \<and> x \<in> U \<and>
               (\<forall>x\<in>U. \<forall>y\<in>U. x \<notin> T \<and> y \<notin> T \<longrightarrow> path_component (S - T) x y)"
           if "x \<in> S" for x
     proof -
@@ -4046,9 +4046,9 @@
       proof (intro exI conjI)
         show "x \<in> ball x r \<inter> affine hull S"
           using \<open>x \<in> S\<close> \<open>r > 0\<close> by (simp add: hull_inc)
-        have "openin (subtopology euclidean (affine hull S)) (ball x r \<inter> affine hull S)"
+        have "openin (top_of_set (affine hull S)) (ball x r \<inter> affine hull S)"
           by (subst inf.commute) (simp add: openin_Int_open)
-        then show "openin (subtopology euclidean S) (ball x r \<inter> affine hull S)"
+        then show "openin (top_of_set S) (ball x r \<inter> affine hull S)"
           by (rule openin_subset_trans [OF _ subS Ssub])
       qed (use * path_component_trans in \<open>auto simp: path_connected_component path_component_of_subset [OF ST]\<close>)
     qed
@@ -4057,7 +4057,7 @@
 
 corollary connected_openin_diff_countable:
   fixes S :: "'a::euclidean_space set"
-  assumes "connected S" and ope: "openin (subtopology euclidean (affine hull S)) S"
+  assumes "connected S" and ope: "openin (top_of_set (affine hull S)) S"
       and "\<not> collinear S" "countable T"
     shows "connected(S - T)"
   by (metis path_connected_imp_connected path_connected_openin_diff_countable [OF assms])
@@ -4074,7 +4074,7 @@
   case False
   show ?thesis
   proof (rule path_connected_openin_diff_countable)
-    show "openin (subtopology euclidean (affine hull S)) S"
+    show "openin (top_of_set (affine hull S)) S"
       by (simp add: assms hull_subset open_subset)
     show "\<not> collinear S"
       using assms False by (simp add: collinear_aff_dim aff_dim_open)
@@ -4329,7 +4329,7 @@
 
 proposition%unimportant homeomorphism_moving_point:
   fixes a :: "'a::euclidean_space"
-  assumes ope: "openin (subtopology euclidean (affine hull S)) S"
+  assumes ope: "openin (top_of_set (affine hull S)) S"
       and "S \<subseteq> T"
       and TS: "T \<subseteq> affine hull S"
       and S: "connected S" "a \<in> S" "b \<in> S"
@@ -4371,7 +4371,7 @@
     then show "bounded {x. \<not> ((f2 \<circ> f1) x = x \<and> (g1 \<circ> g2) x = x)}"
       by (rule bounded_subset) auto
   qed
-  have 3: "\<exists>U. openin (subtopology euclidean S) U \<and>
+  have 3: "\<exists>U. openin (top_of_set S) U \<and>
               d \<in> U \<and>
               (\<forall>x\<in>U.
                   \<exists>f g. homeomorphism T T f g \<and> f d = x \<and>
@@ -4410,7 +4410,7 @@
   assumes K: "finite K" "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
              "pairwise (\<lambda>i j. (x i \<noteq> x j) \<and> (y i \<noteq> y j)) K"
       and "2 \<le> aff_dim S"
-      and ope: "openin (subtopology euclidean (affine hull S)) S"
+      and ope: "openin (top_of_set (affine hull S)) S"
       and "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
   shows "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(x i) = y i) \<and>
                {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
@@ -4449,7 +4449,7 @@
                and hk_sub: "{x. \<not> (h x = x \<and> k x = x)} \<subseteq> S - y ` K"
                and bo_hk:  "bounded {x. \<not> (h x = x \<and> k x = x)}"
   proof (rule homeomorphism_moving_point [of "S - y`K" T "f(x i)" "y i"])
-    show "openin (subtopology euclidean (affine hull (S - y ` K))) (S - y ` K)"
+    show "openin (top_of_set (affine hull (S - y ` K))) (S - y ` K)"
       by (simp add: aff_eq openin_diff finite_imp_closedin image_subset_iff hull_inc insert xyS)
     show "S - y ` K \<subseteq> T"
       using \<open>S \<subseteq> T\<close> by auto
@@ -4499,7 +4499,7 @@
   case False
   then have affS: "affine hull S = UNIV"
     by (simp add: affine_hull_open \<open>open S\<close>)
-  then have ope: "openin (subtopology euclidean (affine hull S)) S"
+  then have ope: "openin (top_of_set (affine hull S)) S"
     using \<open>open S\<close> open_openin by auto
   have "2 \<le> DIM('a)" by (rule 2)
   also have "\<dots> = aff_dim (UNIV :: 'a set)"
@@ -4714,7 +4714,7 @@
       using \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> by auto
     show "homeomorphism T T f' g'"
     proof
-      have clo: "closedin (subtopology euclidean (cbox w z \<union> (T - box w z))) (T - box w z)"
+      have clo: "closedin (top_of_set (cbox w z \<union> (T - box w z))) (T - box w z)"
         by (metis Diff_Diff_Int Diff_subset T closedin_def open_box openin_open_Int topspace_euclidean_subtopology)
       have "continuous_on (cbox w z \<union> (T - box w z)) f'" "continuous_on (cbox w z \<union> (T - box w z)) g'"
         unfolding f'_def g'_def
@@ -4830,14 +4830,14 @@
 
 proposition%unimportant homeomorphism_grouping_points_exists_gen:
   fixes S :: "'a::euclidean_space set"
-  assumes opeU: "openin (subtopology euclidean S) U"
-      and opeS: "openin (subtopology euclidean (affine hull S)) S"
+  assumes opeU: "openin (top_of_set S) U"
+      and opeS: "openin (top_of_set (affine hull S)) S"
       and "U \<noteq> {}" "finite K" "K \<subseteq> S" and S: "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
   obtains f g where "homeomorphism T T f g" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
                     "bounded {x. (\<not> (f x = x \<and> g x = x))}" "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"
 proof (cases "2 \<le> aff_dim S")
   case True
-  have opeU': "openin (subtopology euclidean (affine hull S)) U"
+  have opeU': "openin (top_of_set (affine hull S)) U"
     using opeS opeU openin_trans by blast
   obtain u where "u \<in> U" "u \<in> S"
     using \<open>U \<noteq> {}\<close> opeU openin_imp_subset by fastforce+
@@ -4892,7 +4892,7 @@
       by (meson Topological_Spaces.connected_continuous_image \<open>connected S\<close> homeomorphism_cont1 homeomorphism_of_subsets homhj hull_subset top_greatest)
     have hUS: "h ` U \<subseteq> h ` S"
       by (meson homeomorphism_imp_open_map homeomorphism_of_subsets homhj hull_subset opeS opeU open_UNIV openin_open_eq)
-    have opn: "openin (subtopology euclidean (affine hull S)) U \<Longrightarrow> open (h ` U)" for U
+    have opn: "openin (top_of_set (affine hull S)) U \<Longrightarrow> open (h ` U)" for U
       using homeomorphism_imp_open_map [OF homhj]  by simp
     have "open (h ` U)" "open (h ` S)"
       by (auto intro: opeS opeU openin_trans opn)
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Jordan_Curve.thy
--- a/src/HOL/Analysis/Jordan_Curve.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Jordan_Curve.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -18,7 +18,7 @@
 proof -
   have [simp]: "a \<notin> S" "a \<notin> T" "b \<notin> S" "b \<notin> T"
     by (meson ComplD ccS ccT connected_component_in)+
-  have clo: "closedin (subtopology euclidean (S \<union> T)) S" "closedin (subtopology euclidean (S \<union> T)) T"
+  have clo: "closedin (top_of_set (S \<union> T)) S" "closedin (top_of_set (S \<union> T)) T"
     by (simp_all add: assms closed_subset compact_imp_closed)
   obtain g where contg: "continuous_on S g"
              and g: "\<And>x. x \<in> S \<Longrightarrow> exp (\<i>* of_real (g x)) = (x - a) /\<^sub>R cmod (x - a) / ((x - b) /\<^sub>R cmod (x - b))"
@@ -276,8 +276,8 @@
                      and "arc g" "pathstart g = a" "pathfinish g = b"
                      and pag_sub: "path_image g - {a,b} \<subseteq> middle"
       proof (rule dense_accessible_frontier_point_pairs [OF \<open>open middle\<close> \<open>connected middle\<close>, of "path_image c \<inter> ball a0 (dist a0 b0)" "path_image c \<inter> ball b0 (dist a0 b0)"])
-        show "openin (subtopology euclidean (frontier middle)) (path_image c \<inter> ball a0 (dist a0 b0))"
-             "openin (subtopology euclidean (frontier middle)) (path_image c \<inter> ball b0 (dist a0 b0))"
+        show "openin (top_of_set (frontier middle)) (path_image c \<inter> ball a0 (dist a0 b0))"
+             "openin (top_of_set (frontier middle)) (path_image c \<inter> ball b0 (dist a0 b0))"
           by (simp_all add: \<open>frontier middle = path_image c\<close> openin_open_Int)
         show "path_image c \<inter> ball a0 (dist a0 b0) \<noteq> path_image c \<inter> ball b0 (dist a0 b0)"
           using \<open>a0 \<noteq> b0\<close> \<open>b0 \<in> path_image c\<close> by auto
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Lebesgue_Measure.thy
--- a/src/HOL/Analysis/Lebesgue_Measure.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Lebesgue_Measure.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -1341,11 +1341,11 @@
 qed
 
 lemma lebesgue_openin:
-   "\<lbrakk>openin (subtopology euclidean S) T; S \<in> sets lebesgue\<rbrakk> \<Longrightarrow> T \<in> sets lebesgue"
+   "\<lbrakk>openin (top_of_set S) T; S \<in> sets lebesgue\<rbrakk> \<Longrightarrow> T \<in> sets lebesgue"
   by (metis borel_open openin_open sets.Int sets_completionI_sets sets_lborel)
 
 lemma lebesgue_closedin:
-   "\<lbrakk>closedin (subtopology euclidean S) T; S \<in> sets lebesgue\<rbrakk> \<Longrightarrow> T \<in> sets lebesgue"
+   "\<lbrakk>closedin (top_of_set S) T; S \<in> sets lebesgue\<rbrakk> \<Longrightarrow> T \<in> sets lebesgue"
   by (metis borel_closed closedin_closed sets.Int sets_completionI_sets sets_lborel)
 
 end
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Polytope.thy
--- a/src/HOL/Analysis/Polytope.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Polytope.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -3921,9 +3921,9 @@
       proof (rule Baire [OF \<open>closed U\<close>])
         show "countable {U - C |C. C \<in> \<S> \<and> aff_dim C < aff_dim U}"
           using \<open>finite \<S>\<close> uncountable_infinite by fastforce
-        have "\<And>C. C \<in> \<S> \<Longrightarrow> openin (subtopology euclidean U) (U-C)"
+        have "\<And>C. C \<in> \<S> \<Longrightarrow> openin (top_of_set U) (U-C)"
           by (metis Sup_upper clo closed_limpt closedin_limpt eq openin_diff openin_subtopology_self)
-        then show "openin (subtopology euclidean U) T \<and> U \<subseteq> closure T"
+        then show "openin (top_of_set U) T \<and> U \<subseteq> closure T"
           if "T \<in> {U - C |C. C \<in> \<S> \<and> aff_dim C < aff_dim U}" for T
           using that dense_complement_convex_closed \<open>closed U\<close> \<open>convex U\<close> by auto
       qed
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Product_Topology.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Product_Topology.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -0,0 +1,459 @@
+theory Product_Topology
+imports Function_Topology   
+begin
+
+lemma subset_UnE: (*FIXME MOVE*)
+  assumes "C \<subseteq> A \<union> B"
+    obtains A' B' where "A' \<subseteq> A" "B' \<subseteq> B" "C = A' \<union> B'"
+  by (metis assms Int_Un_distrib inf.order_iff inf_le2)
+
+section \<open>Product Topology\<close> 
+
+subsection\<open>A binary product topology where the two types can be different.\<close>
+
+definition prod_topology :: "'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<times> 'b) topology" where
+ "prod_topology X Y \<equiv> topology (arbitrary union_of (\<lambda>U. U \<in> {S \<times> T |S T. openin X S \<and> openin Y T}))"
+
+lemma open_product_open:
+  assumes "open A"
+  shows "\<exists>\<U>. \<U> \<subseteq> {S \<times> T |S T. open S \<and> open T} \<and> \<Union> \<U> = A"
+proof -
+  obtain f g where *: "\<And>u. u \<in> A \<Longrightarrow> open (f u) \<and> open (g u) \<and> u \<in> (f u) \<times> (g u) \<and> (f u) \<times> (g u) \<subseteq> A"
+    using open_prod_def [of A] assms by metis
+  let ?\<U> = "(\<lambda>u. f u \<times> g u) ` A"
+  show ?thesis
+    by (rule_tac x="?\<U>" in exI) (auto simp: dest: *)
+qed
+
+lemma open_product_open_eq: "(arbitrary union_of (\<lambda>U. \<exists>S T. U = S \<times> T \<and> open S \<and> open T)) = open"
+  by (force simp: union_of_def arbitrary_def intro: open_product_open open_Times)
+
+lemma openin_prod_topology:
+   "openin (prod_topology X Y) = arbitrary union_of (\<lambda>U. U \<in> {S \<times> T |S T. openin X S \<and> openin Y T})"
+  unfolding prod_topology_def
+proof (rule topology_inverse')
+  show "istopology (arbitrary union_of (\<lambda>U. U \<in> {S \<times> T |S T. openin X S \<and> openin Y T}))"
+    apply (rule istopology_base, simp)
+    by (metis openin_Int times_Int_times)
+qed
+
+lemma topspace_prod_topology [simp]:
+   "topspace (prod_topology X Y) = topspace X \<times> topspace Y"
+proof -
+  have "topspace(prod_topology X Y) = \<Union> (Collect (openin (prod_topology X Y)))" (is "_ = ?Z")
+    unfolding topspace_def ..
+  also have "\<dots> = topspace X \<times> topspace Y"
+  proof
+    show "?Z \<subseteq> topspace X \<times> topspace Y"
+      apply (auto simp: openin_prod_topology union_of_def arbitrary_def)
+      using openin_subset by force+
+  next
+    have *: "\<exists>A B. topspace X \<times> topspace Y = A \<times> B \<and> openin X A \<and> openin Y B"
+      by blast
+    show "topspace X \<times> topspace Y \<subseteq> ?Z"
+      apply (rule Union_upper)
+      using * by (simp add: openin_prod_topology arbitrary_union_of_inc)
+  qed
+  finally show ?thesis .
+qed
+
+lemma subtopology_Times:
+  shows "subtopology (prod_topology X Y) (S \<times> T) = prod_topology (subtopology X S) (subtopology Y T)"
+proof -
+  have "((\<lambda>U. \<exists>S T. U = S \<times> T \<and> openin X S \<and> openin Y T) relative_to S \<times> T) =
+        (\<lambda>U. \<exists>S' T'. U = S' \<times> T' \<and> (openin X relative_to S) S' \<and> (openin Y relative_to T) T')"
+    by (auto simp: relative_to_def times_Int_times fun_eq_iff) metis
+  then show ?thesis
+    by (simp add: topology_eq openin_prod_topology arbitrary_union_of_relative_to flip: openin_relative_to)
+qed
+
+lemma prod_topology_subtopology:
+    "prod_topology (subtopology X S) Y = subtopology (prod_topology X Y) (S \<times> topspace Y)"
+    "prod_topology X (subtopology Y T) = subtopology (prod_topology X Y) (topspace X \<times> T)"
+by (auto simp: subtopology_Times)
+
+lemma prod_topology_discrete_topology:
+     "discrete_topology (S \<times> T) = prod_topology (discrete_topology S) (discrete_topology T)"
+  by (auto simp: discrete_topology_unique openin_prod_topology intro: arbitrary_union_of_inc)
+
+lemma prod_topology_euclidean [simp]: "prod_topology euclidean euclidean = euclidean"
+  by (simp add: prod_topology_def open_product_open_eq)
+
+lemma prod_topology_subtopology_eu [simp]:
+  "prod_topology (subtopology euclidean S) (subtopology euclidean T) = subtopology euclidean (S \<times> T)"
+  by (simp add: prod_topology_subtopology subtopology_subtopology times_Int_times)
+
+lemma continuous_map_subtopology_eu [simp]:
+  "continuous_map (subtopology euclidean S) (subtopology euclidean T) h \<longleftrightarrow> continuous_on S h \<and> h ` S \<subseteq> T"
+  apply safe
+  apply (metis continuous_map_closedin_preimage_eq continuous_map_in_subtopology continuous_on_closed order_refl topspace_euclidean_subtopology)
+  apply (simp add: continuous_map_closedin_preimage_eq image_subset_iff)
+  by (metis (no_types, hide_lams) continuous_map_closedin_preimage_eq continuous_map_in_subtopology continuous_on_closed order_refl topspace_euclidean_subtopology)
+
+lemma openin_prod_topology_alt:
+     "openin (prod_topology X Y) S \<longleftrightarrow>
+      (\<forall>x y. (x,y) \<in> S \<longrightarrow> (\<exists>U V. openin X U \<and> openin Y V \<and> x \<in> U \<and> y \<in> V \<and> U \<times> V \<subseteq> S))"
+  apply (auto simp: openin_prod_topology arbitrary_union_of_alt, fastforce)
+  by (metis mem_Sigma_iff)
+
+lemma open_map_fst: "open_map (prod_topology X Y) X fst"
+  unfolding open_map_def openin_prod_topology_alt
+  by (force simp: openin_subopen [of X "fst ` _"] intro: subset_fst_imageI)
+
+lemma open_map_snd: "open_map (prod_topology X Y) Y snd"
+  unfolding open_map_def openin_prod_topology_alt
+  by (force simp: openin_subopen [of Y "snd ` _"] intro: subset_snd_imageI)
+
+lemma openin_Times:
+     "openin (prod_topology X Y) (S \<times> T) \<longleftrightarrow> S = {} \<or> T = {} \<or> openin X S \<and> openin Y T"
+proof (cases "S = {} \<or> T = {}")
+  case False
+  then show ?thesis
+    apply (simp add: openin_prod_topology_alt openin_subopen [of X S] openin_subopen [of Y T] times_subset_iff, safe)
+      apply (meson|force)+
+    done
+qed force
+
+
+lemma closure_of_Times:
+   "(prod_topology X Y) closure_of (S \<times> T) = (X closure_of S) \<times> (Y closure_of T)"  (is "?lhs = ?rhs")
+proof
+  show "?lhs \<subseteq> ?rhs"
+    by (clarsimp simp: closure_of_def openin_prod_topology_alt) blast
+  show "?rhs \<subseteq> ?lhs"
+    by (clarsimp simp: closure_of_def openin_prod_topology_alt) (meson SigmaI subsetD)
+qed
+
+lemma closedin_Times:
+   "closedin (prod_topology X Y) (S \<times> T) \<longleftrightarrow> S = {} \<or> T = {} \<or> closedin X S \<and> closedin Y T"
+  by (auto simp: closure_of_Times times_eq_iff simp flip: closure_of_eq)
+
+lemma continuous_map_pairwise:
+   "continuous_map Z (prod_topology X Y) f \<longleftrightarrow> continuous_map Z X (fst \<circ> f) \<and> continuous_map Z Y (snd \<circ> f)"
+   (is "?lhs = ?rhs")
+proof -
+  let ?g = "fst \<circ> f" and ?h = "snd \<circ> f"
+  have f: "f x = (?g x, ?h x)" for x
+    by auto
+  show ?thesis
+  proof (cases "(\<forall>x \<in> topspace Z. ?g x \<in> topspace X) \<and> (\<forall>x \<in> topspace Z. ?h x \<in> topspace Y)")
+    case True
+    show ?thesis
+    proof safe
+      assume "continuous_map Z (prod_topology X Y) f"
+      then have "openin Z {x \<in> topspace Z. fst (f x) \<in> U}" if "openin X U" for U
+        unfolding continuous_map_def using True that
+        apply clarify
+        apply (drule_tac x="U \<times> topspace Y" in spec)
+        by (simp add: openin_Times mem_Times_iff cong: conj_cong)
+      with True show "continuous_map Z X (fst \<circ> f)"
+        by (auto simp: continuous_map_def)
+    next
+      assume "continuous_map Z (prod_topology X Y) f"
+      then have "openin Z {x \<in> topspace Z. snd (f x) \<in> V}" if "openin Y V" for V
+        unfolding continuous_map_def using True that
+        apply clarify
+        apply (drule_tac x="topspace X \<times> V" in spec)
+        by (simp add: openin_Times mem_Times_iff cong: conj_cong)
+      with True show "continuous_map Z Y (snd \<circ> f)"
+        by (auto simp: continuous_map_def)
+    next
+      assume Z: "continuous_map Z X (fst \<circ> f)" "continuous_map Z Y (snd \<circ> f)"
+      have *: "openin Z {x \<in> topspace Z. f x \<in> W}"
+        if "\<And>w. w \<in> W \<Longrightarrow> \<exists>U V. openin X U \<and> openin Y V \<and> w \<in> U \<times> V \<and> U \<times> V \<subseteq> W" for W
+      proof (subst openin_subopen, clarify)
+        fix x :: "'a"
+        assume "x \<in> topspace Z" and "f x \<in> W"
+        with that [OF \<open>f x \<in> W\<close>]
+        obtain U V where UV: "openin X U" "openin Y V" "f x \<in> U \<times> V" "U \<times> V \<subseteq> W"
+          by auto
+        with Z  UV show "\<exists>T. openin Z T \<and> x \<in> T \<and> T \<subseteq> {x \<in> topspace Z. f x \<in> W}"
+          apply (rule_tac x="{x \<in> topspace Z. ?g x \<in> U} \<inter> {x \<in> topspace Z. ?h x \<in> V}" in exI)
+          apply (auto simp: \<open>x \<in> topspace Z\<close> continuous_map_def)
+          done
+      qed
+      show "continuous_map Z (prod_topology X Y) f"
+        using True by (simp add: continuous_map_def openin_prod_topology_alt mem_Times_iff *)
+    qed
+  qed (auto simp: continuous_map_def)
+qed
+
+lemma continuous_map_paired:
+   "continuous_map Z (prod_topology X Y) (\<lambda>x. (f x,g x)) \<longleftrightarrow> continuous_map Z X f \<and> continuous_map Z Y g"
+  by (simp add: continuous_map_pairwise o_def)
+
+lemma continuous_map_pairedI [continuous_intros]:
+   "\<lbrakk>continuous_map Z X f; continuous_map Z Y g\<rbrakk> \<Longrightarrow> continuous_map Z (prod_topology X Y) (\<lambda>x. (f x,g x))"
+  by (simp add: continuous_map_pairwise o_def)
+
+lemma continuous_map_fst [continuous_intros]: "continuous_map (prod_topology X Y) X fst"
+  using continuous_map_pairwise [of "prod_topology X Y" X Y id]
+  by (simp add: continuous_map_pairwise)
+
+lemma continuous_map_snd [continuous_intros]: "continuous_map (prod_topology X Y) Y snd"
+  using continuous_map_pairwise [of "prod_topology X Y" X Y id]
+  by (simp add: continuous_map_pairwise)
+
+lemma continuous_map_fst_of [continuous_intros]:
+   "continuous_map Z (prod_topology X Y) f \<Longrightarrow> continuous_map Z X (fst \<circ> f)"
+  by (simp add: continuous_map_pairwise)
+
+lemma continuous_map_snd_of [continuous_intros]:
+   "continuous_map Z (prod_topology X Y) f \<Longrightarrow> continuous_map Z Y (snd \<circ> f)"
+  by (simp add: continuous_map_pairwise)
+
+lemma continuous_map_if_iff [simp]: "continuous_map X Y (\<lambda>x. if P then f x else g x) \<longleftrightarrow> continuous_map X Y (if P then f else g)"
+  by simp
+
+lemma continuous_map_if [continuous_intros]: "\<lbrakk>P \<Longrightarrow> continuous_map X Y f; ~P \<Longrightarrow> continuous_map X Y g\<rbrakk>
+      \<Longrightarrow> continuous_map X Y (\<lambda>x. if P then f x else g x)"
+  by simp
+
+lemma continuous_map_subtopology_fst [continuous_intros]: "continuous_map (subtopology (prod_topology X Y) Z) X fst"
+      using continuous_map_from_subtopology continuous_map_fst by force
+
+lemma continuous_map_subtopology_snd [continuous_intros]: "continuous_map (subtopology (prod_topology X Y) Z) Y snd"
+      using continuous_map_from_subtopology continuous_map_snd by force
+
+lemma quotient_map_fst [simp]:
+   "quotient_map(prod_topology X Y) X fst \<longleftrightarrow> (topspace Y = {} \<longrightarrow> topspace X = {})"
+  by (auto simp: continuous_open_quotient_map open_map_fst continuous_map_fst)
+
+lemma quotient_map_snd [simp]:
+   "quotient_map(prod_topology X Y) Y snd \<longleftrightarrow> (topspace X = {} \<longrightarrow> topspace Y = {})"
+  by (auto simp: continuous_open_quotient_map open_map_snd continuous_map_snd)
+
+lemma retraction_map_fst:
+   "retraction_map (prod_topology X Y) X fst \<longleftrightarrow> (topspace Y = {} \<longrightarrow> topspace X = {})"
+proof (cases "topspace Y = {}")
+  case True
+  then show ?thesis
+    using continuous_map_image_subset_topspace
+    by (fastforce simp: retraction_map_def retraction_maps_def continuous_map_fst continuous_map_on_empty)
+next
+  case False
+  have "\<exists>g. continuous_map X (prod_topology X Y) g \<and> (\<forall>x\<in>topspace X. fst (g x) = x)"
+    if y: "y \<in> topspace Y" for y
+    by (rule_tac x="\<lambda>x. (x,y)" in exI) (auto simp: y continuous_map_paired)
+  with False have "retraction_map (prod_topology X Y) X fst"
+    by (fastforce simp: retraction_map_def retraction_maps_def continuous_map_fst)
+  with False show ?thesis
+    by simp
+qed
+
+lemma retraction_map_snd:
+   "retraction_map (prod_topology X Y) Y snd \<longleftrightarrow> (topspace X = {} \<longrightarrow> topspace Y = {})"
+proof (cases "topspace X = {}")
+  case True
+  then show ?thesis
+    using continuous_map_image_subset_topspace
+    by (fastforce simp: retraction_map_def retraction_maps_def continuous_map_fst continuous_map_on_empty)
+next
+  case False
+  have "\<exists>g. continuous_map Y (prod_topology X Y) g \<and> (\<forall>y\<in>topspace Y. snd (g y) = y)"
+    if x: "x \<in> topspace X" for x
+    by (rule_tac x="\<lambda>y. (x,y)" in exI) (auto simp: x continuous_map_paired)
+  with False have "retraction_map (prod_topology X Y) Y snd"
+    by (fastforce simp: retraction_map_def retraction_maps_def continuous_map_snd)
+  with False show ?thesis
+    by simp
+qed
+
+
+lemma continuous_map_of_fst:
+   "continuous_map (prod_topology X Y) Z (f \<circ> fst) \<longleftrightarrow> topspace Y = {} \<or> continuous_map X Z f"
+proof (cases "topspace Y = {}")
+  case True
+  then show ?thesis
+    by (simp add: continuous_map_on_empty)
+next
+  case False
+  then show ?thesis
+    by (simp add: continuous_compose_quotient_map_eq quotient_map_fst)
+qed
+
+lemma continuous_map_of_snd:
+   "continuous_map (prod_topology X Y) Z (f \<circ> snd) \<longleftrightarrow> topspace X = {} \<or> continuous_map Y Z f"
+proof (cases "topspace X = {}")
+  case True
+  then show ?thesis
+    by (simp add: continuous_map_on_empty)
+next
+  case False
+  then show ?thesis
+    by (simp add: continuous_compose_quotient_map_eq quotient_map_snd)
+qed
+
+lemma continuous_map_prod_top:
+   "continuous_map (prod_topology X Y) (prod_topology X' Y') (\<lambda>(x,y). (f x, g y)) \<longleftrightarrow>
+    topspace (prod_topology X Y) = {} \<or> continuous_map X X' f \<and> continuous_map Y Y' g"
+proof (cases "topspace (prod_topology X Y) = {}")
+  case True
+  then show ?thesis
+    by (simp add: continuous_map_on_empty)
+next
+  case False
+  then show ?thesis
+    by (simp add: continuous_map_paired case_prod_unfold continuous_map_of_fst [unfolded o_def] continuous_map_of_snd [unfolded o_def])
+qed
+
+lemma homeomorphic_maps_prod:
+   "homeomorphic_maps (prod_topology X Y) (prod_topology X' Y') (\<lambda>(x,y). (f x, g y)) (\<lambda>(x,y). (f' x, g' y)) \<longleftrightarrow>
+        topspace(prod_topology X Y) = {} \<and>
+        topspace(prod_topology X' Y') = {} \<or>
+        homeomorphic_maps X X' f f' \<and>
+        homeomorphic_maps Y Y' g g'"
+  unfolding homeomorphic_maps_def continuous_map_prod_top
+  by (auto simp: continuous_map_def homeomorphic_maps_def continuous_map_prod_top)
+
+lemma embedding_map_graph:
+   "embedding_map X (prod_topology X Y) (\<lambda>x. (x, f x)) \<longleftrightarrow> continuous_map X Y f"
+    (is "?lhs = ?rhs")
+proof
+  assume L: ?lhs
+  have "snd \<circ> (\<lambda>x. (x, f x)) = f"
+    by force
+  moreover have "continuous_map X Y (snd \<circ> (\<lambda>x. (x, f x)))"
+    using L
+    unfolding embedding_map_def
+    by (meson continuous_map_in_subtopology continuous_map_snd_of homeomorphic_imp_continuous_map)
+  ultimately show ?rhs
+    by simp
+next
+  assume R: ?rhs
+  then show ?lhs
+    unfolding homeomorphic_map_maps embedding_map_def homeomorphic_maps_def
+    by (rule_tac x=fst in exI)
+       (auto simp: continuous_map_in_subtopology continuous_map_paired continuous_map_from_subtopology
+                   continuous_map_fst topspace_subtopology)
+qed
+
+lemma in_prod_topology_closure_of:
+  assumes  "z \<in> (prod_topology X Y) closure_of S"
+  shows "fst z \<in> X closure_of (fst ` S)" "snd z \<in> Y closure_of (snd ` S)"
+  using assms continuous_map_eq_image_closure_subset continuous_map_fst apply fastforce
+  using assms continuous_map_eq_image_closure_subset continuous_map_snd apply fastforce
+  done
+
+
+proposition compact_space_prod_topology:
+   "compact_space(prod_topology X Y) \<longleftrightarrow> topspace(prod_topology X Y) = {} \<or> compact_space X \<and> compact_space Y"
+proof (cases "topspace(prod_topology X Y) = {}")
+  case True
+  then show ?thesis
+    using compact_space_topspace_empty by blast
+next
+  case False
+  then have non_mt: "topspace X \<noteq> {}" "topspace Y \<noteq> {}"
+    by auto
+  have "compact_space X" "compact_space Y" if "compact_space(prod_topology X Y)"
+  proof -
+    have "compactin X (fst ` (topspace X \<times> topspace Y))"
+      by (metis compact_space_def continuous_map_fst image_compactin that topspace_prod_topology)
+    moreover
+    have "compactin Y (snd ` (topspace X \<times> topspace Y))"
+      by (metis compact_space_def continuous_map_snd image_compactin that topspace_prod_topology)
+    ultimately show "compact_space X" "compact_space Y"
+      by (simp_all add: non_mt compact_space_def)
+  qed
+  moreover
+  define \<X> where "\<X> \<equiv> (\<lambda>V. topspace X \<times> V) ` Collect (openin Y)"
+  define \<Y> where "\<Y> \<equiv> (\<lambda>U. U \<times> topspace Y) ` Collect (openin X)"
+  have "compact_space(prod_topology X Y)" if "compact_space X" "compact_space Y"
+  proof (rule Alexander_subbase_alt)
+    show toptop: "topspace X \<times> topspace Y \<subseteq> \<Union>(\<X> \<union> \<Y>)"
+      unfolding \<X>_def \<Y>_def by auto
+    fix \<C> :: "('a \<times> 'b) set set"
+    assume \<C>: "\<C> \<subseteq> \<X> \<union> \<Y>" "topspace X \<times> topspace Y \<subseteq> \<Union>\<C>"
+    then obtain \<X>' \<Y>' where XY: "\<X>' \<subseteq> \<X>" "\<Y>' \<subseteq> \<Y>" and \<C>eq: "\<C> = \<X>' \<union> \<Y>'"
+      using subset_UnE by metis
+    then have sub: "topspace X \<times> topspace Y \<subseteq> \<Union>(\<X>' \<union> \<Y>')"
+      using \<C> by simp
+    obtain \<U> \<V> where \<U>: "\<And>U. U \<in> \<U> \<Longrightarrow> openin X U" "\<Y>' = (\<lambda>U. U \<times> topspace Y) ` \<U>"
+      and \<V>: "\<And>V. V \<in> \<V> \<Longrightarrow> openin Y V" "\<X>' = (\<lambda>V. topspace X \<times> V) ` \<V>"
+      using XY by (clarsimp simp add: \<X>_def \<Y>_def subset_image_iff) (force simp add: subset_iff)
+    have "\<exists>\<D>. finite \<D> \<and> \<D> \<subseteq> \<X>' \<union> \<Y>' \<and> topspace X \<times> topspace Y \<subseteq> \<Union> \<D>"
+    proof -
+      have "topspace X \<subseteq> \<Union>\<U> \<or> topspace Y \<subseteq> \<Union>\<V>"
+        using \<U> \<V> \<C> \<C>eq by auto
+      then have *: "\<exists>\<D>. finite \<D> \<and>
+               (\<forall>x \<in> \<D>. x \<in> (\<lambda>V. topspace X \<times> V) ` \<V> \<or> x \<in> (\<lambda>U. U \<times> topspace Y) ` \<U>) \<and>
+               (topspace X \<times> topspace Y \<subseteq> \<Union>\<D>)"
+      proof
+        assume "topspace X \<subseteq> \<Union>\<U>"
+        with \<open>compact_space X\<close> \<U> obtain \<F> where "finite \<F>" "\<F> \<subseteq> \<U>" "topspace X \<subseteq> \<Union>\<F>"
+          by (meson compact_space_alt)
+        with that show ?thesis
+          by (rule_tac x="(\<lambda>D. D \<times> topspace Y) ` \<F>" in exI) auto
+      next
+        assume "topspace Y \<subseteq> \<Union>\<V>"
+        with \<open>compact_space Y\<close> \<V> obtain \<F> where "finite \<F>" "\<F> \<subseteq> \<V>" "topspace Y \<subseteq> \<Union>\<F>"
+          by (meson compact_space_alt)
+        with that show ?thesis
+          by (rule_tac x="(\<lambda>C. topspace X \<times> C) ` \<F>" in exI) auto
+      qed
+      then show ?thesis
+        using that \<U> \<V> by blast
+    qed
+    then show "\<exists>\<D>. finite \<D> \<and> \<D> \<subseteq> \<C> \<and> topspace X \<times> topspace Y \<subseteq> \<Union> \<D>"
+      using \<C> \<C>eq by blast
+  next
+    have "(finite intersection_of (\<lambda>x. x \<in> \<X> \<or> x \<in> \<Y>) relative_to topspace X \<times> topspace Y)
+           = (\<lambda>U. \<exists>S T. U = S \<times> T \<and> openin X S \<and> openin Y T)"
+      (is "?lhs = ?rhs")
+    proof -
+      have "?rhs U" if "?lhs U" for U
+      proof -
+        have "topspace X \<times> topspace Y \<inter> \<Inter> T \<in> {A \<times> B |A B. A \<in> Collect (openin X) \<and> B \<in> Collect (openin Y)}"
+          if "finite T" "T \<subseteq> \<X> \<union> \<Y>" for T
+          using that
+        proof induction
+          case (insert B \<B>)
+          then show ?case
+            unfolding \<X>_def \<Y>_def
+            apply (simp add: Int_ac subset_eq image_def)
+            apply (metis (no_types) openin_Int openin_topspace times_Int_times)
+            done
+        qed auto
+        then show ?thesis
+          using that
+          by (auto simp: subset_eq  elim!: relative_toE intersection_ofE)
+      qed
+      moreover
+      have "?lhs Z" if Z: "?rhs Z" for Z
+      proof -
+        obtain U V where "Z = U \<times> V" "openin X U" "openin Y V"
+          using Z by blast
+        then have UV: "U \<times> V = (topspace X \<times> topspace Y) \<inter> (U \<times> V)"
+          by (simp add: Sigma_mono inf_absorb2 openin_subset)
+        moreover
+        have "?lhs ((topspace X \<times> topspace Y) \<inter> (U \<times> V))"
+        proof (rule relative_to_inc)
+          show "(finite intersection_of (\<lambda>x. x \<in> \<X> \<or> x \<in> \<Y>)) (U \<times> V)"
+            apply (simp add: intersection_of_def \<X>_def \<Y>_def)
+            apply (rule_tac x="{(U \<times> topspace Y),(topspace X \<times> V)}" in exI)
+            using \<open>openin X U\<close> \<open>openin Y V\<close> openin_subset UV apply (fastforce simp add:)
+            done
+        qed
+        ultimately show ?thesis
+          using \<open>Z = U \<times> V\<close> by auto
+      qed
+      ultimately show ?thesis
+        by meson
+    qed
+    then show "topology (arbitrary union_of (finite intersection_of (\<lambda>x. x \<in> \<X> \<union> \<Y>)
+           relative_to (topspace X \<times> topspace Y))) =
+        prod_topology X Y"
+      by (simp add: prod_topology_def)
+  qed
+  ultimately show ?thesis
+    using False by blast
+qed
+
+
+lemma compactin_Times:
+   "compactin (prod_topology X Y) (S \<times> T) \<longleftrightarrow> S = {} \<or> T = {} \<or> compactin X S \<and> compactin Y T"
+  by (auto simp: compactin_subspace subtopology_Times compact_space_prod_topology)
+
+end
+
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Riemann_Mapping.thy
--- a/src/HOL/Analysis/Riemann_Mapping.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Riemann_Mapping.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -1029,14 +1029,14 @@
           by (metis frontierS C)
         obtain K where "C \<subseteq> K" "compact K"
                    and Ksub: "K \<subseteq> \<Inter>(range X)" and clo: "closed(\<Inter>(range X) - K)"
-        proof (cases "{k. C \<subseteq> k \<and> compact k \<and> openin (subtopology euclidean (\<Inter>(range X))) k} = {}")
+        proof (cases "{k. C \<subseteq> k \<and> compact k \<and> openin (top_of_set (\<Inter>(range X))) k} = {}")
           case True
           then show ?thesis
             using Sura_Bura [OF lcX Cco \<open>compact C\<close>] boC
             by (simp add: True)
         next
           case False
-          then obtain L where "compact L" "C \<subseteq> L" and K: "openin (subtopology euclidean (\<Inter>x. X x)) L"
+          then obtain L where "compact L" "C \<subseteq> L" and K: "openin (top_of_set (\<Inter>x. X x)) L"
             by blast
           show ?thesis
           proof
@@ -1089,9 +1089,9 @@
                 apply safe
                 using DiffI J empty apply auto[1]
                 using closure_subset by blast
-              then have "openin (subtopology euclidean (X j)) (X j \<inter> closure U)"
+              then have "openin (top_of_set (X j)) (X j \<inter> closure U)"
                 by (simp add: openin_open_Int \<open>open U\<close>)
-              moreover have "closedin (subtopology euclidean (X j)) (X j \<inter> closure U)"
+              moreover have "closedin (top_of_set (X j)) (X j \<inter> closure U)"
                 by (simp add: closedin_closed_Int)
               moreover have "X j \<inter> closure U \<noteq> X j"
                 by (metis unboundedX \<open>compact (closure U)\<close> bounded_subset compact_eq_bounded_closed inf.order_iff)
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Starlike.thy
--- a/src/HOL/Analysis/Starlike.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Starlike.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -8,7 +8,7 @@
 chapter \<open>Unsorted\<close>
 
 theory Starlike
-imports Convex_Euclidean_Space
+imports Convex_Euclidean_Space Abstract_Limits
 begin
 
 section \<open>Line Segments\<close>
@@ -1838,7 +1838,7 @@
   assumes "convex S"
   shows "rel_interior (rel_interior S) = rel_interior S"
 proof -
-  have "openin (subtopology euclidean (affine hull (rel_interior S))) (rel_interior S)"
+  have "openin (top_of_set (affine hull (rel_interior S))) (rel_interior S)"
     using openin_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
   then show ?thesis
     using rel_interior_def by auto
@@ -2083,7 +2083,7 @@
   fixes S :: "'n::euclidean_space set"
   shows "closed (rel_frontier S)"
 proof -
-  have *: "closedin (subtopology euclidean (affine hull S)) (closure S - rel_interior S)"
+  have *: "closedin (top_of_set (affine hull S)) (closure S - rel_interior S)"
     by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
   show ?thesis
     apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
@@ -4792,11 +4792,11 @@
 
 lemma separation_normal_local:
   fixes S :: "'a::euclidean_space set"
-  assumes US: "closedin (subtopology euclidean U) S"
-      and UT: "closedin (subtopology euclidean U) T"
+  assumes US: "closedin (top_of_set U) S"
+      and UT: "closedin (top_of_set U) T"
       and "S \<inter> T = {}"
-  obtains S' T' where "openin (subtopology euclidean U) S'"
-                      "openin (subtopology euclidean U) T'"
+  obtains S' T' where "openin (top_of_set U) S'"
+                      "openin (top_of_set U) T'"
                       "S \<subseteq> S'"  "T \<subseteq> T'"  "S' \<inter> T' = {}"
 proof (cases "S = {} \<or> T = {}")
   case True with that show ?thesis
@@ -4810,10 +4810,10 @@
   proof (rule_tac S' = "(U \<inter> f -` {0<..})" and T' = "(U \<inter> f -` {..<0})" in that)
     show "(U \<inter> f -` {0<..}) \<inter> (U \<inter> f -` {..<0}) = {}"
       by auto
-    show "openin (subtopology euclidean U) (U \<inter> f -` {0<..})"
+    show "openin (top_of_set U) (U \<inter> f -` {0<..})"
       by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
   next
-    show "openin (subtopology euclidean U) (U \<inter> f -` {..<0})"
+    show "openin (top_of_set U) (U \<inter> f -` {..<0})"
       by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
   next
     have "S \<subseteq> U" "T \<subseteq> U"
@@ -4993,10 +4993,10 @@
 
 proposition proper_map:
   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
-  assumes "closedin (subtopology euclidean S) K"
+  assumes "closedin (top_of_set S) K"
       and com: "\<And>U. \<lbrakk>U \<subseteq> T; compact U\<rbrakk> \<Longrightarrow> compact (S \<inter> f -` U)"
       and "f ` S \<subseteq> T"
-    shows "closedin (subtopology euclidean T) (f ` K)"
+    shows "closedin (top_of_set T) (f ` K)"
 proof -
   have "K \<subseteq> S"
     using assms closedin_imp_subset by metis
@@ -5221,7 +5221,7 @@
 lemma proper_map_from_compact:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T" and "compact S"
-          "closedin (subtopology euclidean T) K"
+          "closedin (top_of_set T) K"
   shows "compact (S \<inter> f -` K)"
 by (rule closedin_compact [OF \<open>compact S\<close>] continuous_closedin_preimage_gen assms)+
 
@@ -5236,8 +5236,8 @@
 
 lemma closed_map_fst:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
-  assumes "compact T" "closedin (subtopology euclidean (S \<times> T)) c"
-   shows "closedin (subtopology euclidean S) (fst ` c)"
+  assumes "compact T" "closedin (top_of_set (S \<times> T)) c"
+   shows "closedin (top_of_set S) (fst ` c)"
 proof -
   have *: "fst ` (S \<times> T) \<subseteq> S"
     by auto
@@ -5256,8 +5256,8 @@
 
 lemma closed_map_snd:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
-  assumes "compact S" "closedin (subtopology euclidean (S \<times> T)) c"
-   shows "closedin (subtopology euclidean T) (snd ` c)"
+  assumes "compact S" "closedin (top_of_set (S \<times> T)) c"
+   shows "closedin (top_of_set T) (snd ` c)"
 proof -
   have *: "snd ` (S \<times> T) \<subseteq> T"
     by auto
@@ -5267,14 +5267,14 @@
 
 lemma closedin_compact_projection:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
-  assumes "compact S" and clo: "closedin (subtopology euclidean (S \<times> T)) U"
-    shows "closedin (subtopology euclidean T) {y. \<exists>x. x \<in> S \<and> (x, y) \<in> U}"
+  assumes "compact S" and clo: "closedin (top_of_set (S \<times> T)) U"
+    shows "closedin (top_of_set T) {y. \<exists>x. x \<in> S \<and> (x, y) \<in> U}"
 proof -
   have "U \<subseteq> S \<times> T"
     by (metis clo closedin_imp_subset)
   then have "{y. \<exists>x. x \<in> S \<and> (x, y) \<in> U} = snd ` U"
     by force
-  moreover have "closedin (subtopology euclidean T) (snd ` U)"
+  moreover have "closedin (top_of_set T) (snd ` U)"
     by (rule closed_map_snd [OF assms])
   ultimately show ?thesis
     by simp
@@ -5366,14 +5366,14 @@
 
 corollary affine_hull_convex_Int_openin:
   fixes S :: "'a::real_normed_vector set"
-  assumes "convex S" "openin (subtopology euclidean (affine hull S)) T" "S \<inter> T \<noteq> {}"
+  assumes "convex S" "openin (top_of_set (affine hull S)) T" "S \<inter> T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 using assms unfolding openin_open
 by (metis affine_hull_convex_Int_open hull_subset inf.orderE inf_assoc)
 
 corollary affine_hull_openin:
   fixes S :: "'a::real_normed_vector set"
-  assumes "openin (subtopology euclidean (affine hull T)) S" "S \<noteq> {}"
+  assumes "openin (top_of_set (affine hull T)) S" "S \<noteq> {}"
     shows "affine hull S = affine hull T"
 using assms unfolding openin_open
 by (metis affine_affine_hull affine_hull_affine_Int_open hull_hull)
@@ -5396,10 +5396,10 @@
 
 lemma affine_hull_Diff:
   fixes S:: "'a::real_normed_vector set"
-  assumes ope: "openin (subtopology euclidean (affine hull S)) S" and "finite F" "F \<subset> S"
+  assumes ope: "openin (top_of_set (affine hull S)) S" and "finite F" "F \<subset> S"
     shows "affine hull (S - F) = affine hull S"
 proof -
-  have clo: "closedin (subtopology euclidean S) F"
+  have clo: "closedin (top_of_set S) F"
     using assms finite_imp_closedin by auto
   moreover have "S - F \<noteq> {}"
     using assms by auto
@@ -6342,7 +6342,7 @@
 
 lemma aff_dim_openin:
   fixes S :: "'a::euclidean_space set"
-  assumes ope: "openin (subtopology euclidean T) S" and "affine T" "S \<noteq> {}"
+  assumes ope: "openin (top_of_set T) S" and "affine T" "S \<noteq> {}"
   shows "aff_dim S = aff_dim T"
 proof -
   show ?thesis
@@ -6394,7 +6394,7 @@
 
 lemma dim_openin:
   fixes S :: "'a::euclidean_space set"
-  assumes ope: "openin (subtopology euclidean T) S" and "subspace T" "S \<noteq> {}"
+  assumes ope: "openin (top_of_set T) S" and "subspace T" "S \<noteq> {}"
   shows "dim S = dim T"
 proof (rule order_antisym)
   show "dim S \<le> dim T"
@@ -6490,7 +6490,7 @@
 corollary%unimportant dense_complement_openin_affine_hull:
   fixes S :: "'a :: euclidean_space set"
   assumes less: "aff_dim T < aff_dim S"
-      and ope: "openin (subtopology euclidean (affine hull S)) S"
+      and ope: "openin (top_of_set (affine hull S)) S"
     shows "closure(S - T) = closure S"
 proof -
   have "affine hull S - T \<subseteq> affine hull S"
@@ -6802,13 +6802,13 @@
 
 corollary paracompact_closedin:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
-  assumes cin: "closedin (subtopology euclidean U) S"
-      and oin: "\<And>T. T \<in> \<C> \<Longrightarrow> openin (subtopology euclidean U) T"
+  assumes cin: "closedin (top_of_set U) S"
+      and oin: "\<And>T. T \<in> \<C> \<Longrightarrow> openin (top_of_set U) T"
       and "S \<subseteq> \<Union>\<C>"
   obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
-               and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (subtopology euclidean U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
+               and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
                and "\<And>x. x \<in> U
-                       \<Longrightarrow> \<exists>V. openin (subtopology euclidean U) V \<and> x \<in> V \<and>
+                       \<Longrightarrow> \<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and>
                                finite {X. X \<in> \<C>' \<and> (X \<inter> V \<noteq> {})}"
 proof -
   have "\<exists>Z. open Z \<and> (T = U \<inter> Z)" if "T \<in> \<C>" for T
@@ -6830,12 +6830,12 @@
   proof (rule_tac \<C>' = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}" in that)
     show "S \<subseteq> \<Union>?C"
       using \<open>U \<inter> K = S\<close> \<open>U \<subseteq> \<Union>\<D>\<close> K by (blast dest!: subsetD)
-    show "\<And>V. V \<in> ?C \<Longrightarrow> openin (subtopology euclidean U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
+    show "\<And>V. V \<in> ?C \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
       using D1 intF by fastforce
     have *: "{X. (\<exists>V. X = U \<inter> V \<and> V \<in> \<D> \<and> V \<inter> K \<noteq> {}) \<and> X \<inter> (U \<inter> V) \<noteq> {}} \<subseteq>
              (\<lambda>x. U \<inter> x) ` {U \<in> \<D>. U \<inter> V \<noteq> {}}" for V
       by blast
-    show "\<exists>V. openin (subtopology euclidean U) V \<and> x \<in> V \<and> finite {X \<in> ?C. X \<inter> V \<noteq> {}}"
+    show "\<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and> finite {X \<in> ?C. X \<inter> V \<noteq> {}}"
          if "x \<in> U" for x
       using D2 [OF that]
       apply clarify
@@ -6862,7 +6862,7 @@
 lemma continuous_closed_graph_gen:
   fixes T :: "'b::real_normed_vector set"
   assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
-    shows "closedin (subtopology euclidean (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
+    shows "closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
 proof -
   have eq: "((\<lambda>x. Pair x (f x)) ` S) =(S \<times> T \<inter> (\<lambda>z. (f \<circ> fst)z - snd z) -` {0})"
     using fim by auto
@@ -6876,16 +6876,16 @@
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact T" and fim: "f ` S \<subseteq> T"
   shows "continuous_on S f \<longleftrightarrow>
-         closedin (subtopology euclidean (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
+         closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
          (is "?lhs = ?rhs")
 proof -
   have "?lhs" if ?rhs
   proof (clarsimp simp add: continuous_on_closed_gen [OF fim])
     fix U
-    assume U: "closedin (subtopology euclidean T) U"
+    assume U: "closedin (top_of_set T) U"
     have eq: "(S \<inter> f -` U) = fst ` (((\<lambda>x. Pair x (f x)) ` S) \<inter> (S \<times> U))"
       by (force simp: image_iff)
-    show "closedin (subtopology euclidean S) (S \<inter> f -` U)"
+    show "closedin (top_of_set S) (S \<inter> f -` U)"
       by (simp add: U closedin_Int closedin_Times closed_map_fst [OF \<open>compact T\<close>] that eq)
   qed
   with continuous_closed_graph_gen assms show ?thesis by blast
@@ -6907,15 +6907,16 @@
     by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF \<open>compact T\<close> fim])
 
 lemma continuous_on_Un_local_open:
-  assumes opS: "openin (subtopology euclidean (S \<union> T)) S"
-      and opT: "openin (subtopology euclidean (S \<union> T)) T"
+  assumes opS: "openin (top_of_set (S \<union> T)) S"
+      and opT: "openin (top_of_set (S \<union> T)) T"
       and contf: "continuous_on S f" and contg: "continuous_on T f"
     shows "continuous_on (S \<union> T) f"
-using pasting_lemma [of "{S,T}" "S \<union> T" "\<lambda>i. i" "\<lambda>i. f" f] contf contg opS opT by blast
+  using pasting_lemma [of "{S,T}" "top_of_set (S \<union> T)" id euclidean "\<lambda>i. f" f] contf contg opS opT
+  by (simp add: subtopology_subtopology) (metis inf.absorb2 openin_imp_subset)  
 
 lemma continuous_on_cases_local_open:
-  assumes opS: "openin (subtopology euclidean (S \<union> T)) S"
-      and opT: "openin (subtopology euclidean (S \<union> T)) T"
+  assumes opS: "openin (top_of_set (S \<union> T)) S"
+      and opT: "openin (top_of_set (S \<union> T)) T"
       and contf: "continuous_on S f" and contg: "continuous_on T g"
       and fg: "\<And>x. x \<in> S \<and> \<not>P x \<or> x \<in> T \<and> P x \<Longrightarrow> f x = g x"
     shows "continuous_on (S \<union> T) (\<lambda>x. if P x then f x else g x)"
@@ -6927,7 +6928,49 @@
   then show ?thesis
     by (rule continuous_on_Un_local_open [OF opS opT])
 qed
-  
+
+lemma continuous_map_cases_le:
+  assumes contp: "continuous_map X euclideanreal p"
+    and contq: "continuous_map X euclideanreal q"
+    and contf: "continuous_map (subtopology X {x. x \<in> topspace X \<and> p x \<le> q x}) Y f"
+    and contg: "continuous_map (subtopology X {x. x \<in> topspace X \<and> q x \<le> p x}) Y g"
+    and fg: "\<And>x. \<lbrakk>x \<in> topspace X; p x = q x\<rbrakk> \<Longrightarrow> f x = g x"
+  shows "continuous_map X Y (\<lambda>x. if p x \<le> q x then f x else g x)"
+proof -
+  have "continuous_map X Y (\<lambda>x. if q x - p x \<in> {0..} then f x else g x)"
+  proof (rule continuous_map_cases_function)
+    show "continuous_map X euclideanreal (\<lambda>x. q x - p x)"
+      by (intro contp contq continuous_intros)
+    show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of {0..}}) Y f"
+      by (simp add: contf)
+    show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of (topspace euclideanreal - {0..})}) Y g"
+      by (simp add: contg flip: Compl_eq_Diff_UNIV)
+  qed (auto simp: fg)
+  then show ?thesis
+    by simp
+qed
+
+lemma continuous_map_cases_lt:
+  assumes contp: "continuous_map X euclideanreal p"
+    and contq: "continuous_map X euclideanreal q"
+    and contf: "continuous_map (subtopology X {x. x \<in> topspace X \<and> p x \<le> q x}) Y f"
+    and contg: "continuous_map (subtopology X {x. x \<in> topspace X \<and> q x \<le> p x}) Y g"
+    and fg: "\<And>x. \<lbrakk>x \<in> topspace X; p x = q x\<rbrakk> \<Longrightarrow> f x = g x"
+  shows "continuous_map X Y (\<lambda>x. if p x < q x then f x else g x)"
+proof -
+  have "continuous_map X Y (\<lambda>x. if q x - p x \<in> {0<..} then f x else g x)"
+  proof (rule continuous_map_cases_function)
+    show "continuous_map X euclideanreal (\<lambda>x. q x - p x)"
+      by (intro contp contq continuous_intros)
+    show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of {0<..}}) Y f"
+      by (simp add: contf)
+    show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of (topspace euclideanreal - {0<..})}) Y g"
+      by (simp add: contg flip: Compl_eq_Diff_UNIV)
+  qed (auto simp: fg)
+  then show ?thesis
+    by simp
+qed
+
 subsection%unimportant\<open>The union of two collinear segments is another segment\<close>
 
 proposition%unimportant in_convex_hull_exchange:
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -1874,8 +1874,8 @@
   obtain \<B> :: "'a set set"
     where "countable \<B>"
       and "{} \<notin> \<B>"
-      and ope: "\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"
-      and if_ope: "\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
+      and ope: "\<And>C. C \<in> \<B> \<Longrightarrow> openin(top_of_set S) C"
+      and if_ope: "\<And>T. openin(top_of_set S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
     by (meson subset_second_countable)
   then obtain f where f: "\<And>C. C \<in> \<B> \<Longrightarrow> f C \<in> C"
     by (metis equals0I)
@@ -1889,7 +1889,7 @@
     proof (clarsimp simp: closure_approachable)
       fix x and e::real
       assume "x \<in> S" "0 < e"
-      have "openin (subtopology euclidean S) (S \<inter> ball x e)"
+      have "openin (top_of_set S) (S \<inter> ball x e)"
         by (simp add: openin_Int_open)
       with if_ope obtain \<U> where  \<U>: "\<U> \<subseteq> \<B>" "S \<inter> ball x e = \<Union>\<U>"
         by meson
@@ -2324,12 +2324,12 @@
 by (simp add: setdist_eq_0_sing_1)
 
 lemma setdist_eq_0_closedin:
-  shows "\<lbrakk>closedin (subtopology euclidean U) S; x \<in> U\<rbrakk>
+  shows "\<lbrakk>closedin (top_of_set U) S; x \<in> U\<rbrakk>
          \<Longrightarrow> (setdist {x} S = 0 \<longleftrightarrow> S = {} \<or> x \<in> S)"
   by (auto simp: closedin_limpt setdist_eq_0_sing_1 closure_def)
 
 lemma setdist_gt_0_closedin:
-  shows "\<lbrakk>closedin (subtopology euclidean U) S; x \<in> U; S \<noteq> {}; x \<notin> S\<rbrakk>
+  shows "\<lbrakk>closedin (top_of_set U) S; x \<in> U; S \<noteq> {}; x \<notin> S\<rbrakk>
          \<Longrightarrow> setdist {x} S > 0"
   using less_eq_real_def setdist_eq_0_closedin by fastforce
 
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Analysis/Vitali_Covering_Theorem.thy
--- a/src/HOL/Analysis/Vitali_Covering_Theorem.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Analysis/Vitali_Covering_Theorem.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -542,10 +542,10 @@
 next
   assume R [rule_format]: "\<forall>x \<in> S. \<forall>e > 0. ?Q x e"
   let ?\<mu> = "measure lebesgue"
-  have "\<exists>U. openin (subtopology euclidean S) U \<and> z \<in> U \<and> negligible U"
+  have "\<exists>U. openin (top_of_set S) U \<and> z \<in> U \<and> negligible U"
     if "z \<in> S" for z
   proof (intro exI conjI)
-    show "openin (subtopology euclidean S) (S \<inter> ball z 1)"
+    show "openin (top_of_set S) (S \<inter> ball z 1)"
       by (simp add: openin_open_Int)
     show "z \<in> S \<inter> ball z 1"
       using \<open>z \<in> S\<close> by auto
diff -r eddcc7c726f3 -r 4a9167f377b0 src/HOL/Product_Type.thy
--- a/src/HOL/Product_Type.thy	Mon Mar 18 15:35:34 2019 +0000
+++ b/src/HOL/Product_Type.thy	Tue Mar 19 16:14:51 2019 +0000
@@ -1124,7 +1124,7 @@
 lemma vimage_snd: "snd -` A = UNIV \<times> A"
   by auto
 
-lemma insert_times_insert [simp]:
+lemma insert_Times_insert [simp]:
   "insert a A \<times> insert b B = insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
   by blast