(* Author: L C Paulson, University of Cambridge
Material split off from Topology_Euclidean_Space
*)
chapter \<open>Connected Components\<close>
theory Connected
imports
Abstract_Topology_2
begin
subsection\<^marker>\<open>tag unimportant\<close> \<open>Connectedness\<close>
lemma connected_local:
"connected S \<longleftrightarrow>
\<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> {})"
using connected_openin by blast
lemma exists_diff:
fixes P :: "'a set \<Rightarrow> bool"
shows "(\<exists>S. P (- S)) \<longleftrightarrow> (\<exists>S. P S)"
by (metis boolean_algebra_class.boolean_algebra.double_compl)
lemma connected_clopen: "connected S \<longleftrightarrow>
(\<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> {})"
unfolding connected_def openin_open closedin_closed
by (metis double_complement)
then have th0: "connected S \<longleftrightarrow>
\<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
(is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")
unfolding closed_def by metis
have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
(is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
unfolding connected_def openin_open closedin_closed by auto
have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" for e2
proof -
have "?P e2 e1 \<longleftrightarrow> (\<exists>t. closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t \<noteq> S)" for e1
by auto
then show ?thesis
by metis
qed
then show ?thesis
by (simp add: th0 th1)
qed
subsection \<open>Connected components, considered as a connectedness relation or a set\<close>
definition\<^marker>\<open>tag important\<close> "connected_component S x y \<equiv> \<exists>T. connected T \<and> T \<subseteq> S \<and> x \<in> T \<and> y \<in> T"
abbreviation "connected_component_set S x \<equiv> Collect (connected_component S x)"
lemma connected_componentI:
"connected T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> x \<in> T \<Longrightarrow> y \<in> T \<Longrightarrow> connected_component S x y"
by (auto simp: connected_component_def)
lemma connected_component_in: "connected_component S x y \<Longrightarrow> x \<in> S \<and> y \<in> S"
by (auto simp: connected_component_def)
lemma connected_component_refl: "x \<in> S \<Longrightarrow> connected_component S x x"
using connected_component_def connected_sing by blast
lemma connected_component_refl_eq [simp]: "connected_component S x x \<longleftrightarrow> x \<in> S"
using connected_component_in connected_component_refl by blast
lemma connected_component_sym: "connected_component S x y \<Longrightarrow> connected_component S y x"
by (auto simp: connected_component_def)
lemma connected_component_trans:
"connected_component S x y \<Longrightarrow> connected_component S y z \<Longrightarrow> connected_component S x z"
unfolding connected_component_def
by (metis Int_iff Un_iff Un_subset_iff equals0D connected_Un)
lemma connected_component_of_subset:
"connected_component S x y \<Longrightarrow> S \<subseteq> T \<Longrightarrow> connected_component T x y"
by (auto simp: connected_component_def)
lemma connected_component_Union: "connected_component_set S x = \<Union>{T. connected T \<and> x \<in> T \<and> T \<subseteq> S}"
by (auto simp: connected_component_def)
lemma connected_connected_component [iff]: "connected (connected_component_set S x)"
by (auto simp: connected_component_Union intro: connected_Union)
lemma connected_iff_eq_connected_component_set:
"connected S \<longleftrightarrow> (\<forall>x \<in> S. connected_component_set S x = S)"
proof
show "\<forall>x\<in>S. connected_component_set S x = S \<Longrightarrow> connected S"
by (metis connectedI_const connected_connected_component)
qed (auto simp: connected_component_def)
lemma connected_component_subset: "connected_component_set S x \<subseteq> S"
using connected_component_in by blast
lemma connected_component_eq_self: "connected S \<Longrightarrow> x \<in> S \<Longrightarrow> connected_component_set S x = S"
by (simp add: connected_iff_eq_connected_component_set)
lemma connected_iff_connected_component:
"connected S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. connected_component S x y)"
using connected_component_in by (auto simp: connected_iff_eq_connected_component_set)
lemma connected_component_maximal:
"x \<in> T \<Longrightarrow> connected T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> T \<subseteq> (connected_component_set S x)"
using connected_component_eq_self connected_component_of_subset by blast
lemma connected_component_mono:
"S \<subseteq> T \<Longrightarrow> connected_component_set S x \<subseteq> connected_component_set T x"
by (simp add: Collect_mono connected_component_of_subset)
lemma connected_component_eq_empty [simp]: "connected_component_set S x = {} \<longleftrightarrow> x \<notin> S"
using connected_component_refl by (fastforce simp: connected_component_in)
lemma connected_component_set_empty [simp]: "connected_component_set {} x = {}"
using connected_component_eq_empty by blast
lemma connected_component_eq:
"y \<in> connected_component_set S x \<Longrightarrow> (connected_component_set S y = connected_component_set S x)"
by (metis (no_types, lifting)
Collect_cong connected_component_sym connected_component_trans mem_Collect_eq)
lemma closed_connected_component:
assumes S: "closed S"
shows "closed (connected_component_set S x)"
proof (cases "x \<in> S")
case False
then show ?thesis
by (metis connected_component_eq_empty closed_empty)
next
case True
show ?thesis
unfolding closure_eq [symmetric]
proof
show "closure (connected_component_set S x) \<subseteq> connected_component_set S x"
proof (rule connected_component_maximal)
show "x \<in> closure (connected_component_set S x)"
by (simp add: closure_def True)
show "connected (closure (connected_component_set S x))"
by (simp add: connected_imp_connected_closure)
show "closure (connected_component_set S x) \<subseteq> S"
by (simp add: S closure_minimal connected_component_subset)
qed
qed (simp add: closure_subset)
qed
lemma connected_component_disjoint:
"connected_component_set S a \<inter> connected_component_set S b = {} \<longleftrightarrow>
a \<notin> connected_component_set S b"
using connected_component_eq connected_component_sym by fastforce
lemma connected_component_nonoverlap:
"connected_component_set S a \<inter> connected_component_set S b = {} \<longleftrightarrow>
a \<notin> S \<or> b \<notin> S \<or> connected_component_set S a \<noteq> connected_component_set S b"
by (metis connected_component_disjoint connected_component_eq connected_component_eq_empty inf.idem)
lemma connected_component_overlap:
"connected_component_set S a \<inter> connected_component_set S b \<noteq> {} \<longleftrightarrow>
a \<in> S \<and> b \<in> S \<and> connected_component_set S a = connected_component_set S b"
by (auto simp: connected_component_nonoverlap)
lemma connected_component_sym_eq: "connected_component S x y \<longleftrightarrow> connected_component S y x"
using connected_component_sym by blast
lemma connected_component_eq_eq:
"connected_component_set S x = connected_component_set S y \<longleftrightarrow>
x \<notin> S \<and> y \<notin> S \<or> x \<in> S \<and> y \<in> S \<and> connected_component S x y"
by (metis connected_component_eq connected_component_eq_empty connected_component_refl mem_Collect_eq)
lemma connected_iff_connected_component_eq:
"connected S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. connected_component_set S x = connected_component_set S y)"
by (simp add: connected_component_eq_eq connected_iff_connected_component)
lemma connected_component_idemp:
"connected_component_set (connected_component_set S x) x = connected_component_set S x"
proof
show "connected_component_set S x \<subseteq> connected_component_set (connected_component_set S x) x"
by (metis connected_component_eq_empty connected_component_maximal order.refl
connected_component_refl connected_connected_component mem_Collect_eq)
qed (simp add: connected_component_subset)
lemma connected_component_unique:
"\<lbrakk>x \<in> c; c \<subseteq> S; connected c;
\<And>c'. \<lbrakk>x \<in> c'; c' \<subseteq> S; connected c'\<rbrakk> \<Longrightarrow> c' \<subseteq> c\<rbrakk>
\<Longrightarrow> connected_component_set S x = c"
by (simp add: connected_component_maximal connected_component_subset subsetD
subset_antisym)
lemma joinable_connected_component_eq:
"\<lbrakk>connected T; T \<subseteq> S;
connected_component_set S x \<inter> T \<noteq> {};
connected_component_set S y \<inter> T \<noteq> {}\<rbrakk>
\<Longrightarrow> connected_component_set S x = connected_component_set S y"
by (metis (full_types) subsetD connected_component_eq connected_component_maximal disjoint_iff_not_equal)
lemma Union_connected_component: "\<Union>(connected_component_set S ` S) = S"
proof
show "\<Union>(connected_component_set S ` S) \<subseteq> S"
by (simp add: SUP_least connected_component_subset)
qed (use connected_component_refl_eq in force)
lemma complement_connected_component_unions:
"S - connected_component_set S x =
\<Union>(connected_component_set S ` S - {connected_component_set S x})"
(is "?lhs = ?rhs")
proof
show "?lhs \<subseteq> ?rhs"
by (metis Diff_subset Diff_subset_conv Sup_insert Union_connected_component insert_Diff_single)
show "?rhs \<subseteq> ?lhs"
by clarsimp (metis connected_component_eq_eq connected_component_in)
qed
lemma connected_component_intermediate_subset:
"\<lbrakk>connected_component_set U a \<subseteq> T; T \<subseteq> U\<rbrakk>
\<Longrightarrow> connected_component_set T a = connected_component_set U a"
by (metis connected_component_idemp connected_component_mono subset_antisym)
lemma connected_component_homeomorphismI:
assumes "homeomorphism A B f g" "connected_component A x y"
shows "connected_component B (f x) (f y)"
proof -
from assms obtain T where T: "connected T" "T \<subseteq> A" "x \<in> T" "y \<in> T"
unfolding connected_component_def by blast
have "connected (f ` T)" "f ` T \<subseteq> B" "f x \<in> f ` T" "f y \<in> f ` T"
using assms T continuous_on_subset[of A f T]
by (auto intro!: connected_continuous_image simp: homeomorphism_def)
thus ?thesis
unfolding connected_component_def by blast
qed
lemma connected_component_homeomorphism_iff:
assumes "homeomorphism A B f g"
shows "connected_component A x y \<longleftrightarrow> x \<in> A \<and> y \<in> A \<and> connected_component B (f x) (f y)"
by (metis assms connected_component_homeomorphismI connected_component_in homeomorphism_apply1 homeomorphism_sym)
lemma connected_component_set_homeomorphism:
assumes "homeomorphism A B f g" "x \<in> A"
shows "connected_component_set B (f x) = f ` connected_component_set A x"
proof -
have "\<And>y. connected_component B (f x) y
\<Longrightarrow> \<exists>u. u \<in> A \<and> connected_component B (f x) (f u) \<and> y = f u"
using assms by (metis connected_component_in homeomorphism_def image_iff)
with assms show ?thesis
by (auto simp: image_iff connected_component_homeomorphism_iff)
qed
subsection \<open>The set of connected components of a set\<close>
definition\<^marker>\<open>tag important\<close> components:: "'a::topological_space set \<Rightarrow> 'a set set"
where "components S \<equiv> connected_component_set S ` S"
lemma components_iff: "S \<in> components U \<longleftrightarrow> (\<exists>x. x \<in> U \<and> S = connected_component_set U x)"
by (auto simp: components_def)
lemma componentsI: "x \<in> U \<Longrightarrow> connected_component_set U x \<in> components U"
by (auto simp: components_def)
lemma componentsE:
assumes "S \<in> components U"
obtains x where "x \<in> U" "S = connected_component_set U x"
using assms by (auto simp: components_def)
lemma Union_components [simp]: "\<Union>(components U) = U"
by (simp add: Union_connected_component components_def)
lemma pairwise_disjoint_components: "pairwise (\<lambda>X Y. X \<inter> Y = {}) (components U)"
unfolding pairwise_def
by (metis (full_types) components_iff connected_component_nonoverlap)
lemma in_components_nonempty: "C \<in> components S \<Longrightarrow> C \<noteq> {}"
by (metis components_iff connected_component_eq_empty)
lemma in_components_subset: "C \<in> components S \<Longrightarrow> C \<subseteq> S"
using Union_components by blast
lemma in_components_connected: "C \<in> components S \<Longrightarrow> connected C"
by (metis components_iff connected_connected_component)
lemma in_components_maximal:
"C \<in> components S \<longleftrightarrow>
C \<noteq> {} \<and> C \<subseteq> S \<and> connected C \<and> (\<forall>D. D \<noteq> {} \<and> C \<subseteq> D \<and> D \<subseteq> S \<and> connected D \<longrightarrow> D = C)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume L: ?lhs
then have "C \<subseteq> S" "connected C"
by (simp_all add: in_components_subset in_components_connected)
then show ?rhs
by (metis (full_types) L components_iff connected_component_maximal connected_component_refl empty_iff mem_Collect_eq subsetD subset_antisym)
next
show "?rhs \<Longrightarrow> ?lhs"
by (metis bot.extremum componentsI connected_component_maximal connected_component_subset
connected_connected_component order_antisym_conv subset_iff)
qed
lemma joinable_components_eq:
"connected T \<and> T \<subseteq> S \<and> c1 \<in> components S \<and> c2 \<in> components S \<and> c1 \<inter> T \<noteq> {} \<and> c2 \<inter> T \<noteq> {} \<Longrightarrow> c1 = c2"
by (metis (full_types) components_iff joinable_connected_component_eq)
lemma closed_components: "\<lbrakk>closed S; C \<in> components S\<rbrakk> \<Longrightarrow> closed C"
by (metis closed_connected_component components_iff)
lemma components_nonoverlap:
"\<lbrakk>C \<in> components S; C' \<in> components S\<rbrakk> \<Longrightarrow> (C \<inter> C' = {}) \<longleftrightarrow> (C \<noteq> C')"
by (metis (full_types) components_iff connected_component_nonoverlap)
lemma components_eq: "\<lbrakk>C \<in> components S; C' \<in> components S\<rbrakk> \<Longrightarrow> (C = C' \<longleftrightarrow> C \<inter> C' \<noteq> {})"
by (metis components_nonoverlap)
lemma components_eq_empty [simp]: "components S = {} \<longleftrightarrow> S = {}"
by (simp add: components_def)
lemma components_empty [simp]: "components {} = {}"
by simp
lemma connected_eq_connected_components_eq: "connected S \<longleftrightarrow> (\<forall>C \<in> components S. \<forall>C' \<in> components S. C = C')"
by (metis (no_types, opaque_lifting) components_iff connected_component_eq_eq connected_iff_connected_component)
lemma components_eq_sing_iff: "components S = {S} \<longleftrightarrow> connected S \<and> S \<noteq> {}" (is "?lhs \<longleftrightarrow> ?rhs")
proof
show "?rhs \<Longrightarrow> ?lhs"
by (metis components_iff connected_component_eq_self equals0I insert_iff mk_disjoint_insert)
qed (use in_components_connected in fastforce)
lemma components_eq_sing_exists: "(\<exists>a. components S = {a}) \<longleftrightarrow> connected S \<and> S \<noteq> {}"
by (metis Union_components ccpo_Sup_singleton components_eq_sing_iff)
lemma connected_eq_components_subset_sing: "connected S \<longleftrightarrow> components S \<subseteq> {S}"
by (metis components_eq_empty components_eq_sing_iff connected_empty subset_singleton_iff)
lemma connected_eq_components_subset_sing_exists: "connected S \<longleftrightarrow> (\<exists>a. components S \<subseteq> {a})"
by (metis components_eq_sing_exists connected_eq_components_subset_sing subset_singleton_iff)
lemma in_components_self: "S \<in> components S \<longleftrightarrow> connected S \<and> S \<noteq> {}"
by (metis components_empty components_eq_sing_iff empty_iff in_components_connected insertI1)
lemma components_maximal: "\<lbrakk>C \<in> components S; connected T; T \<subseteq> S; C \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow> T \<subseteq> C"
by (metis (lifting) ext Int_Un_eq(4) Int_absorb Un_upper1 bot_eq_sup_iff connected_Un
in_components_maximal sup.mono sup.orderI)
lemma exists_component_superset: "\<lbrakk>T \<subseteq> S; S \<noteq> {}; connected T\<rbrakk> \<Longrightarrow> \<exists>C. C \<in> components S \<and> T \<subseteq> C"
by (meson componentsI connected_component_maximal equals0I subset_eq)
lemma components_intermediate_subset: "\<lbrakk>S \<in> components U; S \<subseteq> T; T \<subseteq> U\<rbrakk> \<Longrightarrow> S \<in> components T"
by (smt (verit, best) dual_order.trans in_components_maximal)
lemma in_components_unions_complement: "C \<in> components S \<Longrightarrow> S - C = \<Union>(components S - {C})"
by (metis complement_connected_component_unions components_def components_iff)
lemma connected_intermediate_closure:
assumes cs: "connected S" and st: "S \<subseteq> T" and ts: "T \<subseteq> closure S"
shows "connected T"
using assms unfolding connected_def
by (smt (verit) Int_assoc inf.absorb_iff2 inf_bot_left open_Int_closure_eq_empty)
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
by (metis closedin_empty)
next
case False
then obtain y where y: "connected_component S x y" and "x \<in> S"
using connected_component_eq_empty by blast
have *: "connected_component_set S x \<subseteq> S \<inter> closure (connected_component_set S x)"
by (auto simp: closure_def connected_component_in)
have **: "x \<in> closure (connected_component_set S x)"
by (simp add: \<open>x \<in> S\<close> closure_def)
have "S \<inter> closure (connected_component_set S x) \<subseteq> connected_component_set S x" if "connected_component S x y"
proof (rule connected_component_maximal)
show "connected (S \<inter> closure (connected_component_set S x))"
using "*" connected_intermediate_closure by blast
qed (use \<open>x \<in> S\<close> ** in auto)
with y * show ?thesis
by (auto simp: closedin_closed)
qed
lemma closedin_component:
"C \<in> components S \<Longrightarrow> closedin (top_of_set S) C"
using closedin_connected_component componentsE by blast
subsection\<^marker>\<open>tag unimportant\<close> \<open>Proving a function is constant on a connected set
by proving that a level set is open\<close>
lemma continuous_levelset_openin_cases:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
shows "connected S \<Longrightarrow> continuous_on S f \<Longrightarrow>
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
lemma continuous_levelset_openin:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
shows "connected S \<Longrightarrow> continuous_on S f \<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
lemma continuous_levelset_open:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
assumes S: "connected S" "continuous_on S f"
and a: "open {x \<in> S. f x = a}" "\<exists>x \<in> S. f x = a"
shows "\<forall>x \<in> S. f x = a"
using a continuous_levelset_openin[OF S, of a, unfolded openin_open]
by fast
subsection\<^marker>\<open>tag unimportant\<close> \<open>Preservation of Connectedness\<close>
lemma homeomorphic_connectedness:
assumes "S homeomorphic T"
shows "connected S \<longleftrightarrow> connected T"
using assms unfolding homeomorphic_def homeomorphism_def by (metis connected_continuous_image)
lemma connected_monotone_quotient_preimage:
assumes "connected T"
and contf: "continuous_on S f" and fim: "f ` S = T"
and opT: "\<And>U. U \<subseteq> T
\<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)
fix U V
assume "open U" and "open V" and "U \<inter> S \<noteq> {}" and "V \<inter> S \<noteq> {}"
and "U \<inter> V \<inter> S = {}" and "S \<subseteq> U \<union> V"
moreover
have disjoint: "f ` (S \<inter> U) \<inter> f ` (S \<inter> V) = {}"
proof -
have False if "y \<in> f ` (S \<inter> U) \<inter> f ` (S \<inter> V)" for y
proof -
have "y \<in> T"
using fim that by blast
show ?thesis
using connectedD [OF connT [OF \<open>y \<in> T\<close>] \<open>open U\<close> \<open>open V\<close>]
\<open>S \<subseteq> U \<union> V\<close> \<open>U \<inter> V \<inter> S = {}\<close> that by fastforce
qed
then show ?thesis by blast
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 (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 (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
then show False
using \<open>connected T\<close> disjoint opeU opeV \<open>U \<inter> S \<noteq> {}\<close> \<open>V \<inter> S \<noteq> {}\<close>
by (auto simp: connected_openin)
qed
lemma connected_open_monotone_preimage:
assumes contf: "continuous_on S f" and fim: "f ` S = T"
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)"
proof -
have contf': "continuous_on (S \<inter> f -` C) f"
by (meson contf continuous_on_subset inf_le1)
have eqC: "f ` (S \<inter> f -` C) = C"
using \<open>C \<subseteq> T\<close> fim by blast
show ?thesis
proof (rule connected_monotone_quotient_preimage [OF \<open>connected C\<close> contf' eqC])
show "connected (S \<inter> f -` C \<inter> f -` {y})" if "y \<in> C" for y
by (metis Int_assoc Int_empty_right Int_insert_right_if1 assms(6) connT in_mono that vimage_Int)
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 (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
lemma connected_closed_monotone_preimage:
assumes contf: "continuous_on S f" and fim: "f ` S = T"
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)"
proof -
have contf': "continuous_on (S \<inter> f -` C) f"
by (meson contf continuous_on_subset inf_le1)
have eqC: "f ` (S \<inter> f -` C) = C"
using \<open>C \<subseteq> T\<close> fim by blast
show ?thesis
proof (rule connected_monotone_quotient_preimage [OF \<open>connected C\<close> contf' eqC])
show "connected (S \<inter> f -` C \<inter> f -` {y})" if "y \<in> C" for y
by (metis Int_assoc Int_empty_right Int_insert_right_if1 \<open>C \<subseteq> T\<close> connT subsetD that vimage_Int)
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 (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
subsection\<open>Lemmas about components\<close>
text \<open>See Newman IV, 3.3 and 3.4.\<close>
lemma connected_Un_clopen_in_complement:
fixes S U :: "'a::metric_space set"
assumes "connected S" "connected U" "S \<subseteq> U"
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;
\<And>x y. \<lbrakk>P x y; S \<subseteq> x\<rbrakk> \<Longrightarrow> False\<rbrakk> \<Longrightarrow> \<not>(\<exists>x y. (P x y))" for P
by metis
show ?thesis
unfolding connected_closedin_eq
proof (rule *)
fix H1 H2
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 (top_of_set S) (S \<inter> H1)"
"closedin (top_of_set S) (S \<inter> H2)"
by (metis Un_upper1 closedin_closed_subset inf_commute)+
moreover have "S \<inter> ((S \<union> T) \<inter> H1) \<union> S \<inter> ((S \<union> T) \<inter> H2) = S"
using H by blast
moreover have "H1 \<inter> (S \<inter> ((S \<union> T) \<inter> H2)) = {}"
using H by blast
ultimately have "S \<inter> H1 = {} \<or> S \<inter> H2 = {}"
by (smt (verit) Int_assoc \<open>connected S\<close> connected_closedin_eq inf_commute inf_sup_absorb)
then show "S \<subseteq> H1 \<or> S \<subseteq> H2"
using H \<open>connected S\<close> unfolding connected_closedin by blast
next
fix H1 H2
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"
by auto
have "T \<subseteq> U"
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 (top_of_set ((U - S) \<union> (S \<union> T))) H2"
proof (rule openin_subtopology_Un)
show "openin (top_of_set (S \<union> T)) H2"
by (metis Diff_cancel H Un_Diff Un_Diff_Int closedin_subset openin_closedin_eq topspace_euclidean_subtopology)
then show "openin (top_of_set (U - S)) H2"
by (meson H2T Un_upper2 opeT openin_subset_trans openin_trans)
qed
moreover have "closedin (top_of_set ((U - S) \<union> (S \<union> T))) H2"
proof (rule closedin_subtopology_Un)
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)
ultimately have H2: "H2 = {} \<or> H2 = U"
using Ueq \<open>connected U\<close> unfolding connected_clopen by metis
then have "H2 \<subseteq> S"
by (metis Diff_partition H Un_Diff_cancel Un_subset_iff \<open>H2 \<subseteq> T\<close> assms(3) inf.orderE opeT openin_imp_subset)
moreover have "T \<subseteq> H2 - S"
by (metis (no_types) H2 H opeT openin_closedin_eq topspace_euclidean_subtopology)
ultimately show False
using H \<open>S \<subseteq> H1\<close> by blast
qed blast
qed
proposition component_diff_connected:
fixes S :: "'a::metric_space set"
assumes "connected S" "connected U" "S \<subseteq> U" and C: "C \<in> components (U - S)"
shows "connected(U - C)"
using \<open>connected S\<close> unfolding connected_closedin_eq not_ex de_Morgan_conj
proof clarify
fix H3 H4
assume clo3: "closedin (top_of_set (U - C)) H3"
and clo4: "closedin (top_of_set (U - C)) H4"
and H34: "H3 \<union> H4 = U - C" "H3 \<inter> H4 = {}" and "H3 \<noteq> {}" and "H4 \<noteq> {}"
and * [rule_format]: "\<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 (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 (top_of_set (U - C)) H3"
by (metis Diff_cancel Un_Diff Un_Diff_Int \<open>H3 \<inter> H4 = {}\<close> \<open>H3 \<union> H4 = U - C\<close> ope4)
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 (top_of_set (U - C)) H4"
by (metis Diff_cancel Diff_triv Int_Un_eq(2) Un_Diff H34 inf_commute ope3)
qed (use clo4 \<open>C \<subseteq> U - S\<close> in auto)
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
moreover have "S \<inter> H4 \<noteq> {}"
using components_maximal [OF C cCH4] \<open>C \<noteq> {}\<close> \<open>C \<subseteq> U - S\<close> \<open>H4 \<noteq> {}\<close> \<open>H4 \<subseteq> U - C\<close> by auto
ultimately show False
using * [of "S \<inter> H3" "S \<inter> H4"] \<open>H3 \<inter> H4 = {}\<close> \<open>C \<subseteq> U - S\<close> \<open>H3 \<union> H4 = U - C\<close> \<open>S \<subseteq> U\<close>
by auto
qed
subsection\<^marker>\<open>tag unimportant\<close>\<open>Constancy of a function from a connected set into a finite, disconnected or discrete set\<close>
text\<open>Still missing: versions for a set that is smaller than R, or countable.\<close>
lemma continuous_disconnected_range_constant:
assumes S: "connected S"
and conf: "continuous_on S f"
and fim: "f \<in> S \<rightarrow> T"
and cct: "\<And>y. y \<in> T \<Longrightarrow> connected_component_set T y = {y}"
shows "f constant_on S"
proof (cases "S = {}")
case True then show ?thesis
by (simp add: constant_on_def)
next
case False
then have "f ` S \<subseteq> {f x}" if "x \<in> S" for x
by (metis PiE S cct connected_component_maximal connected_continuous_image [OF conf] fim image_eqI
image_subset_iff that)
with False show ?thesis
unfolding constant_on_def by blast
qed
text\<open>This proof requires the existence of two separate values of the range type.\<close>
lemma finite_range_constant_imp_connected:
assumes "\<And>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
\<lbrakk>continuous_on S f; finite(f ` S)\<rbrakk> \<Longrightarrow> f constant_on S"
shows "connected S"
proof -
{ fix T 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 "continuous_on (T \<union> U) (\<lambda>x. if x \<in> T then 0 else 1)"
using clt clu tue by (intro continuous_on_cases_local) (auto simp: tus)
then have conif: "continuous_on S (\<lambda>x. if x \<in> T then 0 else 1)"
using tus by blast
have fi: "finite ((\<lambda>x. if x \<in> T then 0 else 1) ` S)"
by (rule finite_subset [of _ "{0,1}"]) auto
have "T = {} \<or> U = {}"
using assms [OF conif fi] tus [symmetric]
by (auto simp: Ball_def constant_on_def) (metis IntI empty_iff one_neq_zero tue)
}
then show ?thesis
by (simp add: connected_closedin_eq)
qed
end