section \<open>Absolute Retracts, Absolute Neighbourhood Retracts and Euclidean Neighbourhood Retracts\<close>
theory Retracts
imports
Brouwer_Fixpoint
Continuous_Extension
begin
text \<open>Absolute retracts (AR), absolute neighbourhood retracts (ANR) and also Euclidean neighbourhood
retracts (ENR). We define AR and ANR by specializing the standard definitions for a set to embedding
in spaces of higher dimension.
John Harrison writes: "This turns out to be sufficient (since any set in \<open>\<real>\<^sup>n\<close> can be
embedded as a closed subset of a convex subset of \<open>\<real>\<^sup>n\<^sup>+\<^sup>1\<close>) to derive the usual
definitions, but we need to split them into two implications because of the lack of type
quantifiers. Then ENR turns out to be equivalent to ANR plus local compactness."\<close>
definition\<^marker>\<open>tag important\<close> AR :: "'a::topological_space set \<Rightarrow> bool" where
"AR S \<equiv> \<forall>U. \<forall>S'::('a * real) set.
S homeomorphic S' \<and> closedin (top_of_set U) S' \<longrightarrow> S' retract_of U"
definition\<^marker>\<open>tag important\<close> ANR :: "'a::topological_space set \<Rightarrow> bool" where
"ANR S \<equiv> \<forall>U. \<forall>S'::('a * real) set.
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\<^marker>\<open>tag important\<close> ENR :: "'a::topological_space set \<Rightarrow> bool" where
"ENR S \<equiv> \<exists>U. open U \<and> S retract_of U"
text \<open>First, show that we do indeed get the "usual" properties of ARs and ANRs.\<close>
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 (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 (top_of_set C) S'"
and hom: "S homeomorphic S'"
by (metis that homeomorphic_closedin_convex)
then have "S' retract_of C"
using \<open>AR S\<close> by (simp add: AR_def)
then obtain r where "S' \<subseteq> C" and contr: "continuous_on C r"
and "r ` C \<subseteq> S'" and rid: "\<And>x. x\<in>S' \<Longrightarrow> r x = x"
by (auto simp: retraction_def retract_of_def)
obtain g h where "homeomorphism S S' g h"
using hom by (force simp: homeomorphic_def)
then have "continuous_on (f ` T) g"
by (meson \<open>f ` T \<subseteq> S\<close> continuous_on_subset homeomorphism_def)
then have contgf: "continuous_on T (g \<circ> f)"
by (metis continuous_on_compose contf)
have gfTC: "(g \<circ> f) ` T \<subseteq> C"
proof -
have "g ` S = S'"
by (metis (no_types) \<open>homeomorphism S S' g h\<close> homeomorphism_def)
with \<open>S' \<subseteq> C\<close> \<open>f ` T \<subseteq> S\<close> show ?thesis by force
qed
obtain f' where f': "continuous_on U f'" "f' ` U \<subseteq> C"
"\<And>x. x \<in> T \<Longrightarrow> f' x = (g \<circ> f) x"
by (metis Dugundji [OF C cloUT contgf gfTC])
show ?thesis
proof (rule_tac g = "h \<circ> r \<circ> f'" in that)
show "continuous_on U (h \<circ> r \<circ> f')"
proof (intro continuous_on_compose f')
show "continuous_on (f' ` U) r"
using continuous_on_subset contr f' by blast
show "continuous_on (r ` f' ` U) h"
using \<open>homeomorphism S S' g h\<close> \<open>f' ` U \<subseteq> C\<close>
unfolding homeomorphism_def
by (metis \<open>r ` C \<subseteq> S'\<close> continuous_on_subset image_mono)
qed
show "(h \<circ> r \<circ> f') ` U \<subseteq> S"
using \<open>homeomorphism S S' g h\<close> \<open>r ` C \<subseteq> S'\<close> \<open>f' ` U \<subseteq> C\<close>
by (fastforce simp: homeomorphism_def)
show "\<And>x. x \<in> T \<Longrightarrow> (h \<circ> r \<circ> f') x = f x"
using \<open>homeomorphism S S' g h\<close> \<open>f ` T \<subseteq> S\<close> f'
by (auto simp: rid homeomorphism_def)
qed
qed
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 (top_of_set U) S'"
shows "S' retract_of U"
proof -
obtain g h where hom: "homeomorphism S S' g h"
using assms by (force simp: homeomorphic_def)
obtain h: "continuous_on S' h" " h ` S' \<subseteq> S"
using hom homeomorphism_def by blast
obtain h' where h': "continuous_on U h'" "h' ` U \<subseteq> S"
and h'h: "\<And>x. x \<in> S' \<Longrightarrow> h' x = h x"
by (blast intro: AR_imp_absolute_extensor [OF \<open>AR S\<close> h clo])
have [simp]: "S' \<subseteq> U" using clo closedin_limpt by blast
show ?thesis
proof (simp add: retraction_def retract_of_def, intro exI conjI)
show "continuous_on U (g \<circ> h')"
by (meson continuous_on_compose continuous_on_subset h' hom homeomorphism_cont1)
show "(g \<circ> h') ` U \<subseteq> S'"
using h' by clarsimp (metis hom subsetD homeomorphism_def imageI)
show "\<forall>x\<in>S'. (g \<circ> h') x = x"
by clarsimp (metis h'h hom homeomorphism_def)
qed
qed
lemma AR_imp_absolute_retract_UNIV:
fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
assumes "AR S" "S homeomorphic S'" "closed S'"
shows "S' retract_of UNIV"
using AR_imp_absolute_retract assms by fastforce
lemma absolute_extensor_imp_AR:
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 (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 (top_of_set U) T"
then obtain g h where hom: "homeomorphism S T g h"
by (force simp: homeomorphic_def)
obtain h: "continuous_on T h" " h ` T \<subseteq> S"
using hom homeomorphism_def by blast
obtain h' where h': "continuous_on U h'" "h' ` U \<subseteq> S"
and h'h: "\<forall>x\<in>T. h' x = h x"
using assms [OF h clo] by blast
have [simp]: "T \<subseteq> U"
using clo closedin_imp_subset by auto
show "T retract_of U"
proof (simp add: retraction_def retract_of_def, intro exI conjI)
show "continuous_on U (g \<circ> h')"
by (meson continuous_on_compose continuous_on_subset h' hom homeomorphism_cont1)
show "(g \<circ> h') ` U \<subseteq> T"
using h' by clarsimp (metis hom subsetD homeomorphism_def imageI)
show "\<forall>x\<in>T. (g \<circ> h') x = x"
by clarsimp (metis h'h hom homeomorphism_def)
qed
qed
lemma AR_eq_absolute_extensor:
fixes S :: "'a::euclidean_space set"
shows "AR S \<longleftrightarrow>
(\<forall>f :: 'a * real \<Rightarrow> 'a.
\<forall>U T. continuous_on T f \<longrightarrow> f ` T \<subseteq> S \<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)))"
by (metis (mono_tags, hide_lams) AR_imp_absolute_extensor absolute_extensor_imp_AR)
lemma AR_imp_retract:
fixes S :: "'a::euclidean_space set"
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
lemma AR_homeomorphic_AR:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "AR T" "S homeomorphic T"
shows "AR S"
unfolding AR_def
by (metis assms AR_imp_absolute_retract homeomorphic_trans [of _ S] homeomorphic_sym)
lemma homeomorphic_AR_iff_AR:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
shows "S homeomorphic T \<Longrightarrow> AR S \<longleftrightarrow> AR T"
by (metis AR_homeomorphic_AR homeomorphic_sym)
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 (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 -
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 (top_of_set C) S'"
and hom: "S homeomorphic S'"
by (metis that homeomorphic_closedin_convex)
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"
by (auto simp: retraction_def retract_of_def)
obtain g h where homgh: "homeomorphism S S' g h"
using hom by (force simp: homeomorphic_def)
have "continuous_on (f ` T) g"
by (meson \<open>f ` T \<subseteq> S\<close> continuous_on_subset homeomorphism_def homgh)
then have contgf: "continuous_on T (g \<circ> f)"
by (intro continuous_on_compose contf)
have gfTC: "(g \<circ> f) ` T \<subseteq> C"
proof -
have "g ` S = S'"
by (metis (no_types) homeomorphism_def homgh)
then show ?thesis
by (metis (no_types) assms(3) cloCS closedin_def image_comp image_mono order.trans topspace_euclidean_subtopology)
qed
obtain f' where contf': "continuous_on U f'"
and "f' ` U \<subseteq> C"
and eq: "\<And>x. x \<in> T \<Longrightarrow> f' x = (g \<circ> f) x"
by (metis Dugundji [OF C cloUT contgf gfTC])
show ?thesis
proof (rule_tac V = "U \<inter> f' -` D" and g = "h \<circ> r \<circ> f'" in that)
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 (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"
proof (rule continuous_on_subset [of S'])
show "continuous_on S' h"
using homeomorphism_def homgh by blast
qed (use \<open>r ` D \<subseteq> S'\<close> in blast)
show "continuous_on (U \<inter> f' -` D) (h \<circ> r \<circ> f')"
by (blast intro: continuous_on_compose conth continuous_on_subset [OF contr] continuous_on_subset [OF contf'])
show "(h \<circ> r \<circ> f') ` (U \<inter> f' -` D) \<subseteq> S"
using \<open>homeomorphism S S' g h\<close> \<open>f' ` U \<subseteq> C\<close> \<open>r ` D \<subseteq> S'\<close>
by (auto simp: homeomorphism_def)
show "\<And>x. x \<in> T \<Longrightarrow> (h \<circ> r \<circ> f') x = f x"
using \<open>homeomorphism S S' g h\<close> \<open>f ` T \<subseteq> S\<close> eq
by (auto simp: rid homeomorphism_def)
qed
qed
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 (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)
obtain h: "continuous_on S' h" " h ` S' \<subseteq> S"
using hom homeomorphism_def by blast
from ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> h clo]
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])
have "S' retract_of V"
proof (simp add: retraction_def retract_of_def, intro exI conjI \<open>S' \<subseteq> V\<close>)
show "continuous_on V (g \<circ> h')"
by (meson continuous_on_compose continuous_on_subset h'(1) h'(2) hom homeomorphism_cont1)
show "(g \<circ> h') ` V \<subseteq> S'"
using h' by clarsimp (metis hom subsetD homeomorphism_def imageI)
show "\<forall>x\<in>S'. (g \<circ> h') x = x"
by clarsimp (metis h'h hom homeomorphism_def)
qed
then show ?thesis
by (rule that [OF opUV])
qed
corollary ANR_imp_absolute_neighbourhood_retract_UNIV:
fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
assumes "ANR S" and hom: "S homeomorphic S'" and clo: "closed S'"
obtains V where "open V" "S' retract_of V"
using ANR_imp_absolute_neighbourhood_retract [OF \<open>ANR S\<close> hom]
by (metis clo closed_closedin open_openin subtopology_UNIV)
corollary neighbourhood_extension_into_ANR:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T" and "ANR T" "closed S"
obtains V g where "S \<subseteq> V" "open V" "continuous_on V g"
"g ` V \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
using ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> contf fim]
by (metis \<open>closed S\<close> closed_closedin open_openin subtopology_UNIV)
lemma absolute_neighbourhood_extensor_imp_ANR:
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 (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 (top_of_set U) T"
then obtain g h where hom: "homeomorphism S T g h"
by (force simp: homeomorphic_def)
obtain h: "continuous_on T h" " h ` T \<subseteq> S"
using hom homeomorphism_def by blast
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
have [simp]: "T \<subseteq> U"
using clo closedin_imp_subset by auto
have "T retract_of V"
proof (simp add: retraction_def retract_of_def, intro exI conjI \<open>T \<subseteq> V\<close>)
show "continuous_on V (g \<circ> h')"
by (meson continuous_on_compose continuous_on_subset h' hom homeomorphism_cont1)
show "(g \<circ> h') ` V \<subseteq> T"
using h' by clarsimp (metis hom subsetD homeomorphism_def imageI)
show "\<forall>x\<in>T. (g \<circ> h') x = x"
by clarsimp (metis h'h hom homeomorphism_def)
qed
then show "\<exists>V. openin (top_of_set U) V \<and> T retract_of V"
using opV by blast
qed
lemma ANR_eq_absolute_neighbourhood_extensor:
fixes S :: "'a::euclidean_space set"
shows "ANR S \<longleftrightarrow>
(\<forall>f :: 'a * real \<Rightarrow> 'a.
\<forall>U T. continuous_on T f \<longrightarrow> f ` T \<subseteq> S \<longrightarrow>
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)))" (is "_ = ?rhs")
proof
assume "ANR S" then show ?rhs
by (metis ANR_imp_absolute_neighbourhood_extensor)
qed (simp add: absolute_neighbourhood_extensor_imp_ANR)
lemma ANR_imp_neighbourhood_retract:
fixes S :: "'a::euclidean_space set"
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 (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"
proof -
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 (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 (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"
proof (rule retract_of_subset [OF S'Z])
show "S' \<subseteq> U - W"
using US' \<open>S' \<subseteq> V\<close> \<open>V \<inter> W = {}\<close> closedin_subset by fastforce
show "U - W \<subseteq> Z"
using Diff_subset_conv \<open>U - Z \<subseteq> W\<close> by blast
qed
ultimately show ?thesis
by (metis Diff_subset_conv Diff_triv Int_Diff_Un Int_absorb1 openin_closedin_eq that topspace_euclidean_subtopology)
qed
lemma ANR_imp_closed_neighbourhood_retract:
fixes S :: "'a::euclidean_space set"
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)
lemma ANR_homeomorphic_ANR:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "ANR T" "S homeomorphic T"
shows "ANR S"
unfolding ANR_def
by (metis assms ANR_imp_absolute_neighbourhood_retract homeomorphic_trans [of _ S] homeomorphic_sym)
lemma homeomorphic_ANR_iff_ANR:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
shows "S homeomorphic T \<Longrightarrow> ANR S \<longleftrightarrow> ANR T"
by (metis ANR_homeomorphic_ANR homeomorphic_sym)
subsection \<open>Analogous properties of ENRs\<close>
lemma ENR_imp_absolute_neighbourhood_retract:
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 (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)
then obtain r where "retraction X S r"
by (auto simp: retract_of_def)
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 (top_of_set U) W"
and WS': "closedin (top_of_set W) S'"
apply (rule locally_compact_closedin_open)
by (meson Int_lower2 assms(3) closedin_imp_subset closedin_subset_trans le_inf_iff openin_open)
obtain f g where hom: "homeomorphism S S' f g"
using assms by (force simp: homeomorphic_def)
have contg: "continuous_on S' g"
using hom homeomorphism_def by blast
moreover have "g ` S' \<subseteq> S" by (metis hom equalityE homeomorphism_def)
ultimately obtain h where conth: "continuous_on W h" and hg: "\<And>x. x \<in> S' \<Longrightarrow> h x = g x"
using Tietze_unbounded [of S' g W] WS' by blast
have "W \<subseteq> U" using UW openin_open by auto
have "S' \<subseteq> W" using WS' closedin_closed by auto
have him: "\<And>x. x \<in> S' \<Longrightarrow> h x \<in> X"
by (metis (no_types) \<open>S retract_of X\<close> hg hom homeomorphism_def image_insert insert_absorb insert_iff retract_of_imp_subset subset_eq)
have "S' retract_of (W \<inter> h -` X)"
proof (simp add: retraction_def retract_of_def, intro exI conjI)
show "S' \<subseteq> W" "S' \<subseteq> h -` X"
using him WS' closedin_imp_subset by blast+
show "continuous_on (W \<inter> h -` X) (f \<circ> r \<circ> h)"
proof (intro continuous_on_compose)
show "continuous_on (W \<inter> h -` X) h"
by (meson conth continuous_on_subset inf_le1)
show "continuous_on (h ` (W \<inter> h -` X)) r"
proof -
have "h ` (W \<inter> h -` X) \<subseteq> X"
by blast
then show "continuous_on (h ` (W \<inter> h -` X)) r"
by (meson \<open>retraction X S r\<close> continuous_on_subset retraction)
qed
show "continuous_on (r ` h ` (W \<inter> h -` X)) f"
proof (rule continuous_on_subset [of S])
show "continuous_on S f"
using hom homeomorphism_def by blast
show "r ` h ` (W \<inter> h -` X) \<subseteq> S"
by (metis \<open>retraction X S r\<close> image_mono image_subset_iff_subset_vimage inf_le2 retraction)
qed
qed
show "(f \<circ> r \<circ> h) ` (W \<inter> h -` X) \<subseteq> S'"
using \<open>retraction X S r\<close> hom
by (auto simp: retraction_def homeomorphism_def)
show "\<forall>x\<in>S'. (f \<circ> r \<circ> h) x = x"
using \<open>retraction X S r\<close> hom by (auto simp: retraction_def homeomorphism_def hg)
qed
then show ?thesis
using UW \<open>open X\<close> conth continuous_openin_preimage_eq openin_trans that by blast
qed
corollary ENR_imp_absolute_neighbourhood_retract_UNIV:
fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
assumes "ENR S" "S homeomorphic S'"
obtains T' where "open T'" "S' retract_of T'"
by (metis ENR_imp_absolute_neighbourhood_retract UNIV_I assms(1) assms(2) open_openin subsetI subtopology_UNIV)
lemma ENR_homeomorphic_ENR:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "ENR T" "S homeomorphic T"
shows "ENR S"
unfolding ENR_def
by (meson ENR_imp_absolute_neighbourhood_retract_UNIV assms homeomorphic_sym)
lemma homeomorphic_ENR_iff_ENR:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "S homeomorphic T"
shows "ENR S \<longleftrightarrow> ENR T"
by (meson ENR_homeomorphic_ENR assms homeomorphic_sym)
lemma ENR_translation:
fixes S :: "'a::euclidean_space set"
shows "ENR(image (\<lambda>x. a + x) S) \<longleftrightarrow> ENR S"
by (meson homeomorphic_sym homeomorphic_translation homeomorphic_ENR_iff_ENR)
lemma ENR_linear_image_eq:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "linear f" "inj f"
shows "ENR (image f S) \<longleftrightarrow> ENR S"
by (meson assms homeomorphic_ENR_iff_ENR linear_homeomorphic_image)
text \<open>Some relations among the concepts. We also relate AR to being a retract of UNIV, which is
often a more convenient proxy in the closed case.\<close>
lemma AR_imp_ANR: "AR S \<Longrightarrow> ANR S"
using ANR_def AR_def by fastforce
lemma ENR_imp_ANR:
fixes S :: "'a::euclidean_space set"
shows "ENR S \<Longrightarrow> ANR S"
by (meson ANR_def ENR_imp_absolute_neighbourhood_retract closedin_imp_subset)
lemma ENR_ANR:
fixes S :: "'a::euclidean_space set"
shows "ENR S \<longleftrightarrow> ANR S \<and> locally compact S"
proof
assume "ENR S"
then have "locally compact S"
using ENR_def open_imp_locally_compact retract_of_locally_compact by auto
then show "ANR S \<and> locally compact S"
using ENR_imp_ANR \<open>ENR S\<close> by blast
next
assume "ANR S \<and> locally compact S"
then have "ANR S" "locally compact S" by auto
then obtain T :: "('a * real) set" where "closed T" "S homeomorphic T"
using locally_compact_homeomorphic_closed
by (metis DIM_prod DIM_real Suc_eq_plus1 lessI)
then show "ENR S"
using \<open>ANR S\<close>
by (meson ANR_imp_absolute_neighbourhood_retract_UNIV ENR_def ENR_homeomorphic_ENR)
qed
lemma AR_ANR:
fixes S :: "'a::euclidean_space set"
shows "AR S \<longleftrightarrow> ANR S \<and> contractible S \<and> S \<noteq> {}"
(is "?lhs = ?rhs")
proof
assume ?lhs
have "aff_dim S < int DIM('a \<times> real)"
using aff_dim_le_DIM [of S] by auto
then obtain C and S' :: "('a * real) set"
where "convex C" "C \<noteq> {}" "closedin (top_of_set C) S'" "S homeomorphic S'"
using homeomorphic_closedin_convex by blast
with \<open>AR S\<close> have "contractible S"
by (meson AR_def convex_imp_contractible homeomorphic_contractible_eq retract_of_contractible)
with \<open>AR S\<close> show ?rhs
using AR_imp_ANR AR_imp_retract by fastforce
next
assume ?rhs
then obtain a and h:: "real \<times> 'a \<Rightarrow> 'a"
where conth: "continuous_on ({0..1} \<times> S) h"
and hS: "h ` ({0..1} \<times> S) \<subseteq> S"
and [simp]: "\<And>x. h(0, x) = x"
and [simp]: "\<And>x. h(1, x) = a"
and "ANR S" "S \<noteq> {}"
by (auto simp: contractible_def homotopic_with_def)
then have "a \<in> S"
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 (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 (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 (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 (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 (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"
and j: "\<And>x. x \<in> W \<Longrightarrow> j x \<in> {0..1}"
and j0: "\<And>x. x \<in> W - V \<Longrightarrow> j x = 1"
and j1: "\<And>x. x \<in> T \<Longrightarrow> j x = 0"
by (rule Urysohn_local [OF WT WWV WVT, of 0 "1::real"]) (auto simp: in_segment)
have Weq: "W = (W - V) \<union> (W - V')"
using \<open>V' \<inter> V = {}\<close> by force
show ?thesis
proof (intro conjI exI)
have *: "continuous_on (W - V') (\<lambda>x. h (j x, g x))"
proof (rule continuous_on_compose2 [OF conth continuous_on_Pair])
show "continuous_on (W - V') j"
by (rule continuous_on_subset [OF contj Diff_subset])
show "continuous_on (W - V') g"
by (metis Diff_subset_conv \<open>W - U \<subseteq> V'\<close> contg continuous_on_subset Un_commute)
show "(\<lambda>x. (j x, g x)) ` (W - V') \<subseteq> {0..1} \<times> S"
using j \<open>g ` U \<subseteq> S\<close> \<open>W - U \<subseteq> V'\<close> by fastforce
qed
show "continuous_on W (\<lambda>x. if x \<in> W - V then a else h (j x, g x))"
proof (subst Weq, rule continuous_on_cases_local)
show "continuous_on (W - V') (\<lambda>x. h (j x, g x))"
using "*" by blast
qed (use WWV WV' Weq j0 j1 in auto)
next
have "h (j (x, y), g (x, y)) \<in> S" if "(x, y) \<in> W" "(x, y) \<in> V" for x y
proof -
have "j(x, y) \<in> {0..1}"
using j that by blast
moreover have "g(x, y) \<in> S"
using \<open>V' \<inter> V = {}\<close> \<open>W - U \<subseteq> V'\<close> \<open>g ` U \<subseteq> S\<close> that by fastforce
ultimately show ?thesis
using hS by blast
qed
with \<open>a \<in> S\<close> \<open>g ` U \<subseteq> S\<close>
show "(\<lambda>x. if x \<in> W - V then a else h (j x, g x)) ` W \<subseteq> S"
by auto
next
show "\<forall>x\<in>T. (if x \<in> W - V then a else h (j x, g x)) = f x"
using \<open>T \<subseteq> V\<close> by (auto simp: j0 j1 gf)
qed
qed
then show ?lhs
by (simp add: AR_eq_absolute_extensor)
qed
lemma ANR_retract_of_ANR:
fixes S :: "'a::euclidean_space set"
assumes "ANR T" and ST: "S retract_of T"
shows "ANR S"
proof (clarsimp simp add: ANR_eq_absolute_neighbourhood_extensor)
fix f::"'a \<times> real \<Rightarrow> 'a" and U W
assume W: "continuous_on W f" "f ` W \<subseteq> S" "closedin (top_of_set U) W"
then obtain r where "S \<subseteq> T" and r: "continuous_on T r" "r ` T \<subseteq> S" "\<forall>x\<in>S. r x = x" "continuous_on W f" "f ` W \<subseteq> S"
"closedin (top_of_set U) W"
by (meson ST retract_of_def retraction_def)
then have "f ` W \<subseteq> T"
by blast
with W obtain V g where V: "W \<subseteq> V" "openin (top_of_set U) V" "continuous_on V g" "g ` V \<subseteq> T" "\<forall>x\<in>W. g x = f x"
by (metis ANR_imp_absolute_neighbourhood_extensor \<open>ANR T\<close>)
with r have "continuous_on V (r \<circ> g) \<and> (r \<circ> g) ` V \<subseteq> S \<and> (\<forall>x\<in>W. (r \<circ> g) x = f x)"
by (metis (no_types, lifting) comp_apply continuous_on_compose continuous_on_subset image_subset_iff)
then show "\<exists>V. W \<subseteq> V \<and> openin (top_of_set U) V \<and> (\<exists>g. continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x\<in>W. g x = f x))"
by (meson V)
qed
lemma AR_retract_of_AR:
fixes S :: "'a::euclidean_space set"
shows "\<lbrakk>AR T; S retract_of T\<rbrakk> \<Longrightarrow> AR S"
using ANR_retract_of_ANR AR_ANR retract_of_contractible by fastforce
lemma ENR_retract_of_ENR:
"\<lbrakk>ENR T; S retract_of T\<rbrakk> \<Longrightarrow> ENR S"
by (meson ENR_def retract_of_trans)
lemma retract_of_UNIV:
fixes S :: "'a::euclidean_space set"
shows "S retract_of UNIV \<longleftrightarrow> AR S \<and> closed S"
by (metis AR_ANR AR_imp_retract ENR_def ENR_imp_ANR closed_UNIV closed_closedin contractible_UNIV empty_not_UNIV open_UNIV retract_of_closed retract_of_contractible retract_of_empty(1) subtopology_UNIV)
lemma compact_AR:
fixes S :: "'a::euclidean_space set"
shows "compact S \<and> AR S \<longleftrightarrow> compact S \<and> S retract_of UNIV"
using compact_imp_closed retract_of_UNIV by blast
text \<open>More properties of ARs, ANRs and ENRs\<close>
lemma not_AR_empty [simp]: "\<not> AR({})"
by (auto simp: AR_def)
lemma ENR_empty [simp]: "ENR {}"
by (simp add: ENR_def)
lemma ANR_empty [simp]: "ANR ({} :: 'a::euclidean_space set)"
by (simp add: ENR_imp_ANR)
lemma convex_imp_AR:
fixes S :: "'a::euclidean_space set"
shows "\<lbrakk>convex S; S \<noteq> {}\<rbrakk> \<Longrightarrow> AR S"
by (metis (mono_tags, lifting) Dugundji absolute_extensor_imp_AR)
lemma convex_imp_ANR:
fixes S :: "'a::euclidean_space set"
shows "convex S \<Longrightarrow> ANR S"
using ANR_empty AR_imp_ANR convex_imp_AR by blast
lemma ENR_convex_closed:
fixes S :: "'a::euclidean_space set"
shows "\<lbrakk>closed S; convex S\<rbrakk> \<Longrightarrow> ENR S"
using ENR_def ENR_empty convex_imp_AR retract_of_UNIV by blast
lemma AR_UNIV [simp]: "AR (UNIV :: 'a::euclidean_space set)"
using retract_of_UNIV by auto
lemma ANR_UNIV [simp]: "ANR (UNIV :: 'a::euclidean_space set)"
by (simp add: AR_imp_ANR)
lemma ENR_UNIV [simp]:"ENR UNIV"
using ENR_def by blast
lemma AR_singleton:
fixes a :: "'a::euclidean_space"
shows "AR {a}"
using retract_of_UNIV by blast
lemma ANR_singleton:
fixes a :: "'a::euclidean_space"
shows "ANR {a}"
by (simp add: AR_imp_ANR AR_singleton)
lemma ENR_singleton: "ENR {a}"
using ENR_def by blast
text \<open>ARs closed under union\<close>
lemma AR_closed_Un_local_aux:
fixes U :: "'a::euclidean_space set"
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 -
have "S \<inter> T \<noteq> {}"
using assms AR_def by fastforce
have "S \<subseteq> U" "T \<subseteq> U"
using assms by (auto simp: closedin_imp_subset)
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 (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 (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'"
using S'_def \<open>S \<subseteq> U\<close> setdist_sing_in_set by fastforce
have "T \<subseteq> T'"
using T'_def \<open>T \<subseteq> U\<close> setdist_sing_in_set by fastforce
have "S \<inter> T \<subseteq> W" "W \<subseteq> U"
using \<open>S \<subseteq> U\<close> by (auto simp: W_def setdist_sing_in_set)
have "(S \<inter> T) retract_of W"
proof (rule AR_imp_absolute_retract [OF \<open>AR(S \<inter> T)\<close>])
show "S \<inter> T homeomorphic S \<inter> T"
by (simp add: homeomorphic_refl)
show "closedin (top_of_set W) (S \<inter> T)"
by (meson \<open>S \<inter> T \<subseteq> W\<close> \<open>W \<subseteq> U\<close> assms closedin_Int closedin_subset_trans)
qed
then obtain r0
where "S \<inter> T \<subseteq> W" and contr0: "continuous_on W r0"
and "r0 ` W \<subseteq> S \<inter> T"
and r0 [simp]: "\<And>x. x \<in> S \<inter> T \<Longrightarrow> r0 x = x"
by (auto simp: retract_of_def retraction_def)
have ST: "x \<in> W \<Longrightarrow> x \<in> S \<longleftrightarrow> x \<in> T" for x
using setdist_eq_0_closedin \<open>S \<inter> T \<noteq> {}\<close> assms
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 (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 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 (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 rim: "r ` (W \<union> S) \<subseteq> S" "r ` (W \<union> T) \<subseteq> T"
using \<open>r0 ` W \<subseteq> S \<inter> T\<close> r_def by auto
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"
proof (rule AR_imp_absolute_extensor [OF \<open>AR S\<close> _ _ cloUWS])
show "continuous_on (W \<union> S) r"
using continuous_on_subset contr sup_assoc by blast
qed (use rim in auto)
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"
proof (rule AR_imp_absolute_extensor [OF \<open>AR T\<close> _ _ cloUWT])
show "continuous_on (W \<union> T) r"
using continuous_on_subset contr sup_assoc by blast
qed (use rim in auto)
have U: "U = S' \<union> T'"
by (force simp: S'_def T'_def)
have cont: "continuous_on U (\<lambda>x. if x \<in> S' then g x else h x)"
unfolding U
apply (rule continuous_on_cases_local)
using US' UT' \<open>S' \<inter> T' = W\<close> \<open>U = S' \<union> T'\<close>
contg conth continuous_on_subset geqr heqr by auto
have UST: "(\<lambda>x. if x \<in> S' then g x else h x) ` U \<subseteq> S \<union> T"
using \<open>g ` U \<subseteq> S\<close> \<open>h ` U \<subseteq> T\<close> by auto
show ?thesis
apply (simp add: retract_of_def retraction_def \<open>S \<subseteq> U\<close> \<open>T \<subseteq> U\<close>)
apply (rule_tac x="\<lambda>x. if x \<in> S' then g x else h x" in exI)
using ST UST \<open>S \<subseteq> S'\<close> \<open>S' \<inter> T' = W\<close> \<open>T \<subseteq> T'\<close> cont geqr heqr r_def by auto
qed
lemma AR_closed_Un_local:
fixes S :: "'a::euclidean_space set"
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 (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 (top_of_set U) (C \<inter> g -` S)"
by (metis STS continuous_on_imp_closedin hom homeomorphism_def closedin_trans [OF _ UC])
have UT: "closedin (top_of_set U) (C \<inter> g -` T)"
by (metis STT continuous_on_closed hom homeomorphism_def closedin_trans [OF _ UC])
have "homeomorphism (C \<inter> g -` S) S g f"
using hom
apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
apply (rule_tac x="f x" in image_eqI, auto)
done
then have ARS: "AR (C \<inter> g -` S)"
using \<open>AR S\<close> homeomorphic_AR_iff_AR homeomorphic_def by blast
have "homeomorphism (C \<inter> g -` T) T g f"
using hom
apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
apply (rule_tac x="f x" in image_eqI, auto)
done
then have ART: "AR (C \<inter> g -` T)"
using \<open>AR T\<close> homeomorphic_AR_iff_AR homeomorphic_def by blast
have "homeomorphism (C \<inter> g -` S \<inter> (C \<inter> g -` T)) (S \<inter> T) g f"
using hom
apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
apply (rule_tac x="f x" in image_eqI, auto)
done
then have ARI: "AR ((C \<inter> g -` S) \<inter> (C \<inter> g -` T))"
using \<open>AR (S \<inter> T)\<close> homeomorphic_AR_iff_AR homeomorphic_def by blast
have "C = (C \<inter> g -` S) \<union> (C \<inter> g -` T)"
using hom by (auto simp: homeomorphism_def)
then show ?thesis
by (metis AR_closed_Un_local_aux [OF US UT ARS ART ARI])
qed
then show ?thesis
by (force simp: AR_def)
qed
corollary AR_closed_Un:
fixes S :: "'a::euclidean_space set"
shows "\<lbrakk>closed S; closed T; AR S; AR T; AR (S \<inter> T)\<rbrakk> \<Longrightarrow> AR (S \<union> T)"
by (metis AR_closed_Un_local_aux closed_closedin retract_of_UNIV subtopology_UNIV)
text \<open>ANRs closed under union\<close>
lemma ANR_closed_Un_local_aux:
fixes U :: "'a::euclidean_space set"
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 (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)
next
case False
then have [simp]: "S \<noteq> {}" "T \<noteq> {}" by auto
have "S \<subseteq> U" "T \<subseteq> U"
using assms by (auto simp: closedin_imp_subset)
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 (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 (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'"
using S'_def \<open>S \<subseteq> U\<close> setdist_sing_in_set by fastforce
have "T \<subseteq> T'"
using T'_def \<open>T \<subseteq> U\<close> setdist_sing_in_set by fastforce
have "S' \<union> T' = U"
by (auto simp: S'_def T'_def)
have "W \<subseteq> S'"
by (simp add: Collect_mono S'_def W_def)
have "W \<subseteq> T'"
by (simp add: Collect_mono T'_def W_def)
have ST_W: "S \<inter> T \<subseteq> W" and "W \<subseteq> U"
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 (top_of_set U) W"
using closedin_Int cloUS' cloUT' by blast
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"
by (meson ANR_imp_closed_neighbourhood_retract ST_W US UT \<open>W \<subseteq> U\<close> \<open>ANR(S \<inter> T)\<close> closedin_Int closedin_subset_trans)
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"
using \<open>W \<subseteq> U\<close> cloWW0 closedin_subset by fastforce
obtain r0
where "S \<inter> T \<subseteq> W0" and contr0: "continuous_on W0 r0" and "r0 ` W0 \<subseteq> S \<inter> T"
and r0 [simp]: "\<And>x. x \<in> S \<inter> T \<Longrightarrow> r0 x = x"
using ret by (force simp: retract_of_def retraction_def)
have ST: "x \<in> W \<Longrightarrow> x \<in> S \<longleftrightarrow> x \<in> T" for x
using assms by (auto simp: W_def setdist_sing_in_set dest!: setdist_eq_0_closedin)
define r where "r \<equiv> \<lambda>x. if x \<in> W0 then r0 x else x"
have "r ` (W0 \<union> S) \<subseteq> S" "r ` (W0 \<union> T) \<subseteq> T"
using \<open>r0 ` W0 \<subseteq> S \<inter> T\<close> r_def by auto
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 (top_of_set (W0 \<union> (S \<union> T))) W0"
using closedin_subset_trans [of U]
by (metis le_sup_iff order_refl cloWW0 cloUW closedin_trans \<open>W0 \<subseteq> U\<close> \<open>S \<subseteq> U\<close> \<open>T \<subseteq> U\<close>)
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 (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 (top_of_set S') W1"
and "g ` W1 \<subseteq> S" and geqr: "\<And>x. x \<in> W0 \<union> S \<Longrightarrow> g x = r x"
proof (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> _ \<open>r ` (W0 \<union> S) \<subseteq> S\<close> cloS'WS])
show "continuous_on (W0 \<union> S) r"
using continuous_on_subset contr sup_assoc by blast
qed auto
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 (top_of_set T') W2"
and "h ` W2 \<subseteq> T" and heqr: "\<And>x. x \<in> W0 \<union> T \<Longrightarrow> h x = r x"
proof (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> _ \<open>r ` (W0 \<union> T) \<subseteq> T\<close> cloT'WT])
show "continuous_on (W0 \<union> T) r"
using continuous_on_subset contr sup_assoc by blast
qed auto
have "S' \<inter> T' = W"
by (force simp: S'_def T'_def W_def)
obtain O1 O2 where O12: "open O1" "W1 = S' \<inter> O1" "open O2" "W2 = T' \<inter> O2"
using opeSW1 opeSW2 by (force simp: openin_open)
show ?thesis
proof
have eq: "W1 - (W - U0) \<union> (W2 - (W - U0))
= ((U - T') \<inter> O1 \<union> (U - S') \<inter> O2 \<union> U \<inter> O1 \<inter> O2) - (W - U0)" (is "?WW1 \<union> ?WW2 = ?rhs")
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 (top_of_set U) (?WW1 \<union> ?WW2)"
by (simp add: eq \<open>open O1\<close> \<open>open O2\<close> cloUS' cloUT' cloUW closedin_diff opeUU0 openin_Int_open openin_Un openin_diff)
obtain SU' where "closed SU'" "S' = U \<inter> SU'"
using cloUS' by (auto simp add: closedin_closed)
moreover have "?WW1 = (?WW1 \<union> ?WW2) \<inter> SU'"
using \<open>S' = U \<inter> SU'\<close> \<open>W1 = S' \<inter> O1\<close> \<open>S' \<union> T' = U\<close> \<open>W2 = T' \<inter> O2\<close> \<open>S' \<inter> T' = W\<close> \<open>W0 \<union> S \<subseteq> W1\<close> U0
by auto
ultimately have cloW1: "closedin (top_of_set (W1 - (W - U0) \<union> (W2 - (W - U0)))) (W1 - (W - U0))"
by (metis closedin_closed_Int)
obtain TU' where "closed TU'" "T' = U \<inter> TU'"
using cloUT' by (auto simp add: closedin_closed)
moreover have "?WW2 = (?WW1 \<union> ?WW2) \<inter> TU'"
using \<open>T' = U \<inter> TU'\<close> \<open>W1 = S' \<inter> O1\<close> \<open>S' \<union> T' = U\<close> \<open>W2 = T' \<inter> O2\<close> \<open>S' \<inter> T' = W\<close> \<open>W0 \<union> T \<subseteq> W2\<close> U0
by auto
ultimately have cloW2: "closedin (top_of_set (?WW1 \<union> ?WW2)) ?WW2"
by (metis closedin_closed_Int)
let ?gh = "\<lambda>x. if x \<in> S' then g x else h x"
have "\<exists>r. continuous_on (?WW1 \<union> ?WW2) r \<and> r ` (?WW1 \<union> ?WW2) \<subseteq> S \<union> T \<and> (\<forall>x\<in>S \<union> T. r x = x)"
proof (intro exI conjI)
show "\<forall>x\<in>S \<union> T. ?gh x = x"
using ST \<open>S' \<inter> T' = W\<close> geqr heqr O12
by (metis Int_iff Un_iff \<open>W0 \<union> S \<subseteq> W1\<close> \<open>W0 \<union> T \<subseteq> W2\<close> r0 r_def sup.order_iff)
have "\<And>x. x \<in> ?WW1 \<and> x \<notin> S' \<or> x \<in> ?WW2 \<and> x \<in> S' \<Longrightarrow> g x = h x"
using O12
by (metis (full_types) DiffD1 DiffD2 DiffI IntE IntI U0(2) UnCI \<open>S' \<inter> T' = W\<close> geqr heqr in_mono)
then show "continuous_on (?WW1 \<union> ?WW2) ?gh"
using continuous_on_cases_local [OF cloW1 cloW2 continuous_on_subset [OF contg] continuous_on_subset [OF conth]]
by simp
show "?gh ` (?WW1 \<union> ?WW2) \<subseteq> S \<union> T"
using \<open>W1 = S' \<inter> O1\<close> \<open>W2 = T' \<inter> O2\<close> \<open>S' \<inter> T' = W\<close> \<open>g ` W1 \<subseteq> S\<close> \<open>h ` W2 \<subseteq> T\<close> \<open>U0 \<inter> W \<subseteq> W0\<close> \<open>W0 \<union> S \<subseteq> W1\<close>
by (auto simp add: image_subset_iff)
qed
then show "S \<union> T retract_of ?WW1 \<union> ?WW2"
using \<open>W0 \<union> S \<subseteq> W1\<close> \<open>W0 \<union> T \<subseteq> W2\<close> ST opeUU0 U0
by (auto simp: retract_of_def retraction_def)
qed
qed
lemma ANR_closed_Un_local:
fixes S :: "'a::euclidean_space set"
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 (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 (top_of_set U) (C \<inter> g -` S)"
by (metis STS UC closedin_trans continuous_on_imp_closedin hom homeomorphism_def)
have UT: "closedin (top_of_set U) (C \<inter> g -` T)"
by (metis STT UC closedin_trans continuous_on_imp_closedin hom homeomorphism_def)
have "homeomorphism (C \<inter> g -` S) S g f"
using hom
apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
by (rule_tac x="f x" in image_eqI, auto)
then have ANRS: "ANR (C \<inter> g -` S)"
using \<open>ANR S\<close> homeomorphic_ANR_iff_ANR homeomorphic_def by blast
have "homeomorphism (C \<inter> g -` T) T g f"
using hom apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
by (rule_tac x="f x" in image_eqI, auto)
then have ANRT: "ANR (C \<inter> g -` T)"
using \<open>ANR T\<close> homeomorphic_ANR_iff_ANR homeomorphic_def by blast
have "homeomorphism (C \<inter> g -` S \<inter> (C \<inter> g -` T)) (S \<inter> T) g f"
using hom
apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
by (rule_tac x="f x" in image_eqI, auto)
then have ANRI: "ANR ((C \<inter> g -` S) \<inter> (C \<inter> g -` T))"
using \<open>ANR (S \<inter> T)\<close> homeomorphic_ANR_iff_ANR homeomorphic_def by blast
have "C = (C \<inter> g -` S) \<union> (C \<inter> g -` T)"
using hom by (auto simp: homeomorphism_def)
then show ?thesis
by (metis ANR_closed_Un_local_aux [OF US UT ANRS ANRT ANRI])
qed
then show ?thesis
by (auto simp: ANR_def)
qed
corollary ANR_closed_Un:
fixes S :: "'a::euclidean_space set"
shows "\<lbrakk>closed S; closed T; ANR S; ANR T; ANR (S \<inter> T)\<rbrakk> \<Longrightarrow> ANR (S \<union> T)"
by (simp add: ANR_closed_Un_local closedin_def diff_eq open_Compl openin_open_Int)
lemma ANR_openin:
fixes S :: "'a::euclidean_space set"
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 (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 (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"
using ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> contf \<open>f ` C \<subseteq> T\<close> cloUC] fim by auto
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 (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])
show "g ` (W \<inter> g -` S) \<subseteq> S"
using gim by blast
show "\<forall>x\<in>C. g x = f x"
using geq by blast
qed
qed
lemma ENR_openin:
fixes S :: "'a::euclidean_space set"
assumes "ENR T" "openin (top_of_set T) S"
shows "ENR S"
by (meson ANR_openin ENR_ANR assms locally_open_subset)
lemma ANR_neighborhood_retract:
fixes S :: "'a::euclidean_space set"
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 (top_of_set U) T"
shows "ENR S"
using ENR_openin ENR_retract_of_ENR assms by blast
lemma ANR_rel_interior:
fixes S :: "'a::euclidean_space set"
shows "ANR S \<Longrightarrow> ANR(rel_interior S)"
by (blast intro: ANR_openin openin_set_rel_interior)
lemma ANR_delete:
fixes S :: "'a::euclidean_space set"
shows "ANR S \<Longrightarrow> ANR(S - {a})"
by (blast intro: ANR_openin openin_delete openin_subtopology_self)
lemma ENR_rel_interior:
fixes S :: "'a::euclidean_space set"
shows "ENR S \<Longrightarrow> ENR(rel_interior S)"
by (blast intro: ENR_openin openin_set_rel_interior)
lemma ENR_delete:
fixes S :: "'a::euclidean_space set"
shows "ENR S \<Longrightarrow> ENR(S - {a})"
by (blast intro: ENR_openin openin_delete openin_subtopology_self)
lemma open_imp_ENR: "open S \<Longrightarrow> ENR S"
using ENR_def by blast
lemma open_imp_ANR:
fixes S :: "'a::euclidean_space set"
shows "open S \<Longrightarrow> ANR S"
by (simp add: ENR_imp_ANR open_imp_ENR)
lemma ANR_ball [iff]:
fixes a :: "'a::euclidean_space"
shows "ANR(ball a r)"
by (simp add: convex_imp_ANR)
lemma ENR_ball [iff]: "ENR(ball a r)"
by (simp add: open_imp_ENR)
lemma AR_ball [simp]:
fixes a :: "'a::euclidean_space"
shows "AR(ball a r) \<longleftrightarrow> 0 < r"
by (auto simp: AR_ANR convex_imp_contractible)
lemma ANR_cball [iff]:
fixes a :: "'a::euclidean_space"
shows "ANR(cball a r)"
by (simp add: convex_imp_ANR)
lemma ENR_cball:
fixes a :: "'a::euclidean_space"
shows "ENR(cball a r)"
using ENR_convex_closed by blast
lemma AR_cball [simp]:
fixes a :: "'a::euclidean_space"
shows "AR(cball a r) \<longleftrightarrow> 0 \<le> r"
by (auto simp: AR_ANR convex_imp_contractible)
lemma ANR_box [iff]:
fixes a :: "'a::euclidean_space"
shows "ANR(cbox a b)" "ANR(box a b)"
by (auto simp: convex_imp_ANR open_imp_ANR)
lemma ENR_box [iff]:
fixes a :: "'a::euclidean_space"
shows "ENR(cbox a b)" "ENR(box a b)"
by (simp_all add: ENR_convex_closed closed_cbox open_box open_imp_ENR)
lemma AR_box [simp]:
"AR(cbox a b) \<longleftrightarrow> cbox a b \<noteq> {}" "AR(box a b) \<longleftrightarrow> box a b \<noteq> {}"
by (auto simp: AR_ANR convex_imp_contractible)
lemma ANR_interior:
fixes S :: "'a::euclidean_space set"
shows "ANR(interior S)"
by (simp add: open_imp_ANR)
lemma ENR_interior:
fixes S :: "'a::euclidean_space set"
shows "ENR(interior S)"
by (simp add: open_imp_ENR)
lemma AR_imp_contractible:
fixes S :: "'a::euclidean_space set"
shows "AR S \<Longrightarrow> contractible S"
by (simp add: AR_ANR)
lemma ENR_imp_locally_compact:
fixes S :: "'a::euclidean_space set"
shows "ENR S \<Longrightarrow> locally compact S"
by (simp add: ENR_ANR)
lemma ANR_imp_locally_path_connected:
fixes S :: "'a::euclidean_space set"
assumes "ANR S"
shows "locally path_connected S"
proof -
obtain U and T :: "('a \<times> real) set"
where "convex U" "U \<noteq> {}"
and UT: "closedin (top_of_set U) T" and "S homeomorphic T"
proof (rule homeomorphic_closedin_convex)
show "aff_dim S < int DIM('a \<times> real)"
using aff_dim_le_DIM [of S] by auto
qed auto
then have "locally path_connected T"
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 (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
obtain Ta where "(openin (top_of_set U) Ta \<and> T retract_of Ta)"
using ANR_def UT \<open>S homeomorphic T\<close> assms by moura
then show ?thesis
using S \<open>U \<noteq> {}\<close> by blast
qed
lemma ANR_imp_locally_connected:
fixes S :: "'a::euclidean_space set"
assumes "ANR S"
shows "locally connected S"
using locally_path_connected_imp_locally_connected ANR_imp_locally_path_connected assms by auto
lemma AR_imp_locally_path_connected:
fixes S :: "'a::euclidean_space set"
assumes "AR S"
shows "locally path_connected S"
by (simp add: ANR_imp_locally_path_connected AR_imp_ANR assms)
lemma AR_imp_locally_connected:
fixes S :: "'a::euclidean_space set"
assumes "AR S"
shows "locally connected S"
using ANR_imp_locally_connected AR_ANR assms by blast
lemma ENR_imp_locally_path_connected:
fixes S :: "'a::euclidean_space set"
assumes "ENR S"
shows "locally path_connected S"
by (simp add: ANR_imp_locally_path_connected ENR_imp_ANR assms)
lemma ENR_imp_locally_connected:
fixes S :: "'a::euclidean_space set"
assumes "ENR S"
shows "locally connected S"
using ANR_imp_locally_connected ENR_ANR assms by blast
lemma ANR_Times:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "ANR S" "ANR T" shows "ANR(S \<times> T)"
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 (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 (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"
proof (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> contf1 _ cloUC])
show "(fst \<circ> f) ` C \<subseteq> S"
using fim by auto
qed auto
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 (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"
proof (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> contf2 _ cloUC])
show "(snd \<circ> f) ` C \<subseteq> T"
using fim by auto
qed auto
show "\<exists>V g. C \<subseteq> 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 (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)
show "(\<lambda>x. (g x, h x)) ` (W1 \<inter> W2) \<subseteq> S \<times> T"
using gim him by blast
show "(\<forall>x\<in>C. (g x, h x) = f x)"
using geq heq by auto
qed
qed
lemma AR_Times:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "AR S" "AR T" shows "AR(S \<times> T)"
using assms by (simp add: AR_ANR ANR_Times contractible_Times)
(* Unused and requires ordered_euclidean_space
subsection\<^marker>\<open>tag unimportant\<close>\<open>Retracts and intervals in ordered euclidean space\<close>
lemma ANR_interval [iff]:
fixes a :: "'a::ordered_euclidean_space"
shows "ANR{a..b}"
by (simp add: interval_cbox)
lemma ENR_interval [iff]:
fixes a :: "'a::ordered_euclidean_space"
shows "ENR{a..b}"
by (auto simp: interval_cbox)
*)
subsection \<open>More advanced properties of ANRs and ENRs\<close>
lemma ENR_rel_frontier_convex:
fixes S :: "'a::euclidean_space set"
assumes "bounded S" "convex S"
shows "ENR(rel_frontier S)"
proof (cases "S = {}")
case True then show ?thesis
by simp
next
case False
with assms have "rel_interior S \<noteq> {}"
by (simp add: rel_interior_eq_empty)
then obtain a where a: "a \<in> rel_interior S"
by auto
have ahS: "affine hull S - {a} \<subseteq> {x. closest_point (affine hull S) x \<noteq> a}"
by (auto simp: closest_point_self)
have "rel_frontier S retract_of affine hull S - {a}"
by (simp add: assms a rel_frontier_retract_of_punctured_affine_hull)
also have "\<dots> retract_of {x. closest_point (affine hull S) x \<noteq> a}"
unfolding retract_of_def retraction_def ahS
apply (rule_tac x="closest_point (affine hull S)" in exI)
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 (top_of_set UNIV) (UNIV \<inter> closest_point (affine hull S) -` (- {a}))"
by (intro continuous_openin_preimage_gen) (auto simp: False affine_imp_convex continuous_on_closest_point)
ultimately show ?thesis
by (meson ENR_convex_closed ENR_delete ENR_retract_of_ENR \<open>rel_frontier S retract_of affine hull S - {a}\<close>
closed_affine_hull convex_affine_hull)
qed
lemma ANR_rel_frontier_convex:
fixes S :: "'a::euclidean_space set"
assumes "bounded S" "convex S"
shows "ANR(rel_frontier S)"
by (simp add: ENR_imp_ANR ENR_rel_frontier_convex assms)
lemma ENR_closedin_Un_local:
fixes S :: "'a::euclidean_space set"
shows "\<lbrakk>ENR S; ENR T; ENR(S \<inter> T);
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)
lemma ENR_closed_Un:
fixes S :: "'a::euclidean_space set"
shows "\<lbrakk>closed S; closed T; ENR S; ENR T; ENR(S \<inter> T)\<rbrakk> \<Longrightarrow> ENR(S \<union> T)"
by (auto simp: closed_subset ENR_closedin_Un_local)
lemma absolute_retract_Un:
fixes S :: "'a::euclidean_space set"
shows "\<lbrakk>S retract_of UNIV; T retract_of UNIV; (S \<inter> T) retract_of UNIV\<rbrakk>
\<Longrightarrow> (S \<union> T) retract_of UNIV"
by (meson AR_closed_Un_local_aux closed_subset retract_of_UNIV retract_of_imp_subset)
lemma retract_from_Un_Int:
fixes S :: "'a::euclidean_space set"
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 -
obtain r where r: "continuous_on T r" "r ` T \<subseteq> S \<inter> T" "\<forall>x\<in>S \<inter> T. r x = x"
using Int by (auto simp: retraction_def retract_of_def)
have "S retract_of S \<union> T"
unfolding retraction_def retract_of_def
proof (intro exI conjI)
show "continuous_on (S \<union> T) (\<lambda>x. if x \<in> S then x else r x)"
using r by (intro continuous_on_cases_local [OF clS clT]) auto
qed (use r in auto)
also have "\<dots> retract_of U"
by (rule Un)
finally show ?thesis .
qed
lemma AR_from_Un_Int_local:
fixes S :: "'a::euclidean_space set"
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"
by (meson AR_imp_retract AR_retract_of_AR Un assms closedin_closed_subset local.Int
retract_from_Un_Int retract_of_refl sup_ge2)
lemma AR_from_Un_Int_local':
fixes S :: "'a::euclidean_space set"
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)
lemma AR_from_Un_Int:
fixes S :: "'a::euclidean_space set"
assumes clo: "closed S" "closed T" and Un: "AR(S \<union> T)" and Int: "AR(S \<inter> T)"
shows "AR S"
by (metis AR_from_Un_Int_local [OF _ _ Un Int] Un_commute clo closed_closedin closedin_closed_subset inf_sup_absorb subtopology_UNIV top_greatest)
lemma ANR_from_Un_Int_local:
fixes S :: "'a::euclidean_space set"
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 (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"
by (auto simp: retraction_def retract_of_def)
have Vsub: "V \<subseteq> S \<union> T"
by (meson ope openin_contains_cball)
have Vsup: "S \<inter> T \<subseteq> V"
by (simp add: retract_of_imp_subset ret)
then have eq: "S \<union> V = ((S \<union> T) - T) \<union> V"
by auto
have eq': "S \<union> V = S \<union> (V \<inter> T)"
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 (top_of_set (S \<union> V \<inter> T)) S"
using clS closedin_subset_trans inf.boundedE by blast
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)
qed (use req continuous_on_id in auto)
with rim have "S retract_of S \<union> V"
unfolding retraction_def retract_of_def using eq' by fastforce
then show ?thesis
using ANR_neighborhood_retract [OF Un]
using \<open>S \<union> V = S \<union> T - T \<union> V\<close> clT ope by fastforce
qed
lemma ANR_from_Un_Int:
fixes S :: "'a::euclidean_space set"
assumes clo: "closed S" "closed T" and Un: "ANR(S \<union> T)" and Int: "ANR(S \<inter> T)"
shows "ANR S"
by (metis ANR_from_Un_Int_local [OF _ _ Un Int] Un_commute clo closed_closedin closedin_closed_subset inf_sup_absorb subtopology_UNIV top_greatest)
lemma ANR_finite_Union_convex_closed:
fixes \<T> :: "'a::euclidean_space set set"
assumes \<T>: "finite \<T>" and clo: "\<And>C. C \<in> \<T> \<Longrightarrow> closed C" and con: "\<And>C. C \<in> \<T> \<Longrightarrow> convex C"
shows "ANR(\<Union>\<T>)"
proof -
have "ANR(\<Union>\<T>)" if "card \<T> < n" for n
using assms that
proof (induction n arbitrary: \<T>)
case 0 then show ?case by simp
next
case (Suc n)
have "ANR(\<Union>\<U>)" if "finite \<U>" "\<U> \<subseteq> \<T>" for \<U>
using that
proof (induction \<U>)
case empty
then show ?case by simp
next
case (insert C \<U>)
have "ANR (C \<union> \<Union>\<U>)"
proof (rule ANR_closed_Un)
show "ANR (C \<inter> \<Union>\<U>)"
unfolding Int_Union
proof (rule Suc)
show "finite ((\<inter>) C ` \<U>)"
by (simp add: insert.hyps(1))
show "\<And>Ca. Ca \<in> (\<inter>) C ` \<U> \<Longrightarrow> closed Ca"
by (metis (no_types, hide_lams) Suc.prems(2) closed_Int subsetD imageE insert.prems insertI1 insertI2)
show "\<And>Ca. Ca \<in> (\<inter>) C ` \<U> \<Longrightarrow> convex Ca"
by (metis (mono_tags, lifting) Suc.prems(3) convex_Int imageE insert.prems insert_subset subsetCE)
show "card ((\<inter>) C ` \<U>) < n"
proof -
have "card \<T> \<le> n"
by (meson Suc.prems(4) not_less not_less_eq)
then show ?thesis
by (metis Suc.prems(1) card_image_le card_seteq insert.hyps insert.prems insert_subset le_trans not_less)
qed
qed
show "closed (\<Union>\<U>)"
using Suc.prems(2) insert.hyps(1) insert.prems by blast
qed (use Suc.prems convex_imp_ANR insert.prems insert.IH in auto)
then show ?case
by simp
qed
then show ?case
using Suc.prems(1) by blast
qed
then show ?thesis
by blast
qed
lemma finite_imp_ANR:
fixes S :: "'a::euclidean_space set"
assumes "finite S"
shows "ANR S"
proof -
have "ANR(\<Union>x \<in> S. {x})"
by (blast intro: ANR_finite_Union_convex_closed assms)
then show ?thesis
by simp
qed
lemma ANR_insert:
fixes S :: "'a::euclidean_space set"
assumes "ANR S" "closed S"
shows "ANR(insert a S)"
by (metis ANR_closed_Un ANR_empty ANR_singleton Diff_disjoint Diff_insert_absorb assms closed_singleton insert_absorb insert_is_Un)
lemma ANR_path_component_ANR:
fixes S :: "'a::euclidean_space set"
shows "ANR S \<Longrightarrow> ANR(path_component_set S x)"
using ANR_imp_locally_path_connected ANR_openin openin_path_component_locally_path_connected by blast
lemma ANR_connected_component_ANR:
fixes S :: "'a::euclidean_space set"
shows "ANR S \<Longrightarrow> ANR(connected_component_set S x)"
by (metis ANR_openin openin_connected_component_locally_connected ANR_imp_locally_connected)
lemma ANR_component_ANR:
fixes S :: "'a::euclidean_space set"
assumes "ANR S" "c \<in> components S"
shows "ANR c"
by (metis ANR_connected_component_ANR assms componentsE)
subsection\<open>Original ANR material, now for ENRs\<close>
lemma ENR_bounded:
fixes S :: "'a::euclidean_space set"
assumes "bounded S"
shows "ENR S \<longleftrightarrow> (\<exists>U. open U \<and> bounded U \<and> S retract_of U)"
(is "?lhs = ?rhs")
proof
obtain r where "0 < r" and r: "S \<subseteq> ball 0 r"
using bounded_subset_ballD assms by blast
assume ?lhs
then show ?rhs
by (meson ENR_def Elementary_Metric_Spaces.open_ball bounded_Int bounded_ball inf_le2 le_inf_iff
open_Int r retract_of_imp_subset retract_of_subset)
next
assume ?rhs
then show ?lhs
using ENR_def by blast
qed
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 (top_of_set U) S'"
shows "S' retract_of U"
proof -
have "AR T"
by (simp add: assms convex_imp_AR)
then have "AR S"
using AR_retract_of_AR assms by auto
then show ?thesis
using assms AR_imp_absolute_retract by metis
qed
lemma absolute_retract_imp_AR:
fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
assumes "S retract_of UNIV" "S homeomorphic S'" "closed S'"
shows "S' retract_of UNIV"
using AR_imp_absolute_retract_UNIV assms retract_of_UNIV by blast
lemma homeomorphic_compact_arness:
fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
assumes "S homeomorphic S'"
shows "compact S \<and> S retract_of UNIV \<longleftrightarrow> compact S' \<and> S' retract_of UNIV"
using assms homeomorphic_compactness
by (metis compact_AR homeomorphic_AR_iff_AR)
lemma absolute_retract_from_Un_Int:
fixes S :: "'a::euclidean_space set"
assumes "(S \<union> T) retract_of UNIV" "(S \<inter> T) retract_of UNIV" "closed S" "closed T"
shows "S retract_of UNIV"
using AR_from_Un_Int assms retract_of_UNIV by auto
lemma ENR_from_Un_Int_gen:
fixes S :: "'a::euclidean_space set"
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"
by (meson ANR_from_Un_Int_local ANR_imp_neighbourhood_retract ENR_ANR ENR_neighborhood_retract assms)
lemma ENR_from_Un_Int:
fixes S :: "'a::euclidean_space set"
assumes "closed S" "closed T" "ENR(S \<union> T)" "ENR(S \<inter> T)"
shows "ENR S"
by (meson ENR_from_Un_Int_gen assms closed_subset sup_ge1 sup_ge2)
lemma ENR_finite_Union_convex_closed:
fixes \<T> :: "'a::euclidean_space set set"
assumes \<T>: "finite \<T>" and clo: "\<And>C. C \<in> \<T> \<Longrightarrow> closed C" and con: "\<And>C. C \<in> \<T> \<Longrightarrow> convex C"
shows "ENR(\<Union> \<T>)"
by (simp add: ENR_ANR ANR_finite_Union_convex_closed \<T> clo closed_Union closed_imp_locally_compact con)
lemma finite_imp_ENR:
fixes S :: "'a::euclidean_space set"
shows "finite S \<Longrightarrow> ENR S"
by (simp add: ENR_ANR finite_imp_ANR finite_imp_closed closed_imp_locally_compact)
lemma ENR_insert:
fixes S :: "'a::euclidean_space set"
assumes "closed S" "ENR S"
shows "ENR(insert a S)"
proof -
have "ENR ({a} \<union> S)"
by (metis ANR_insert ENR_ANR Un_commute Un_insert_right assms closed_imp_locally_compact closed_insert sup_bot_right)
then show ?thesis
by auto
qed
lemma ENR_path_component_ENR:
fixes S :: "'a::euclidean_space set"
assumes "ENR S"
shows "ENR(path_component_set S x)"
by (metis ANR_imp_locally_path_connected ENR_empty ENR_imp_ANR ENR_openin assms
locally_path_connected_2 openin_subtopology_self path_component_eq_empty)
(*UNUSED
lemma ENR_Times:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "ENR S" "ENR T" shows "ENR(S \<times> T)"
using assms apply (simp add: ENR_ANR ANR_Times)
thm locally_compact_Times
oops
SIMP_TAC[ENR_ANR; ANR_PCROSS; LOCALLY_COMPACT_PCROSS]);;
*)
subsection\<open>Finally, spheres are ANRs and ENRs\<close>
lemma absolute_retract_homeomorphic_convex_compact:
fixes S :: "'a::euclidean_space set" and U :: "'b::euclidean_space set"
assumes "S homeomorphic U" "S \<noteq> {}" "S \<subseteq> T" "convex U" "compact U"
shows "S retract_of T"
by (metis UNIV_I assms compact_AR convex_imp_AR homeomorphic_AR_iff_AR homeomorphic_compactness homeomorphic_empty(1) retract_of_subset subsetI)
lemma frontier_retract_of_punctured_universe:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "bounded S" "a \<in> interior S"
shows "(frontier S) retract_of (- {a})"
using rel_frontier_retract_of_punctured_affine_hull
by (metis Compl_eq_Diff_UNIV affine_hull_nonempty_interior assms empty_iff rel_frontier_frontier rel_interior_nonempty_interior)
lemma sphere_retract_of_punctured_universe_gen:
fixes a :: "'a::euclidean_space"
assumes "b \<in> ball a r"
shows "sphere a r retract_of (- {b})"
proof -
have "frontier (cball a r) retract_of (- {b})"
using assms frontier_retract_of_punctured_universe interior_cball by blast
then show ?thesis
by simp
qed
lemma sphere_retract_of_punctured_universe:
fixes a :: "'a::euclidean_space"
assumes "0 < r"
shows "sphere a r retract_of (- {a})"
by (simp add: assms sphere_retract_of_punctured_universe_gen)
lemma ENR_sphere:
fixes a :: "'a::euclidean_space"
shows "ENR(sphere a r)"
proof (cases "0 < r")
case True
then have "sphere a r retract_of -{a}"
by (simp add: sphere_retract_of_punctured_universe)
with open_delete show ?thesis
by (auto simp: ENR_def)
next
case False
then show ?thesis
using finite_imp_ENR
by (metis finite_insert infinite_imp_nonempty less_linear sphere_eq_empty sphere_trivial)
qed
corollary\<^marker>\<open>tag unimportant\<close> ANR_sphere:
fixes a :: "'a::euclidean_space"
shows "ANR(sphere a r)"
by (simp add: ENR_imp_ANR ENR_sphere)
subsection\<open>Spheres are connected, etc\<close>
lemma locally_path_connected_sphere_gen:
fixes S :: "'a::euclidean_space set"
assumes "bounded S" and "convex S"
shows "locally path_connected (rel_frontier S)"
proof (cases "rel_interior S = {}")
case True
with assms show ?thesis
by (simp add: rel_interior_eq_empty)
next
case False
then obtain a where a: "a \<in> rel_interior S"
by blast
show ?thesis
proof (rule retract_of_locally_path_connected)
show "locally path_connected (affine hull S - {a})"
by (meson convex_affine_hull convex_imp_locally_path_connected locally_open_subset openin_delete openin_subtopology_self)
show "rel_frontier S retract_of affine hull S - {a}"
using a assms rel_frontier_retract_of_punctured_affine_hull by blast
qed
qed
lemma locally_connected_sphere_gen:
fixes S :: "'a::euclidean_space set"
assumes "bounded S" and "convex S"
shows "locally connected (rel_frontier S)"
by (simp add: ANR_imp_locally_connected ANR_rel_frontier_convex assms)
lemma locally_path_connected_sphere:
fixes a :: "'a::euclidean_space"
shows "locally path_connected (sphere a r)"
using ENR_imp_locally_path_connected ENR_sphere by blast
lemma locally_connected_sphere:
fixes a :: "'a::euclidean_space"
shows "locally connected(sphere a r)"
using ANR_imp_locally_connected ANR_sphere by blast
subsection\<open>Borsuk homotopy extension theorem\<close>
text\<open>It's only this late so we can use the concept of retraction,
saying that the domain sets or range set are ENRs.\<close>
theorem Borsuk_homotopy_extension_homotopic:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
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"
and "homotopic_with_canon (\<lambda>x. True) S U f g"
obtains g' where "homotopic_with_canon (\<lambda>x. True) T U f g'"
"continuous_on T g'" "image g' T \<subseteq> U"
"\<And>x. x \<in> S \<Longrightarrow> g' x = g x"
proof -
have "S \<subseteq> T" using assms closedin_imp_subset by blast
obtain h where conth: "continuous_on ({0..1} \<times> S) h"
and him: "h ` ({0..1} \<times> S) \<subseteq> U"
and [simp]: "\<And>x. h(0, x) = f x" "\<And>x. h(1::real, x) = g x"
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 (top_of_set ({0..1} \<times> T)) ({0::real} \<times> T)"
by (simp add: Abstract_Topology.closedin_Times)
moreover have cloT1S: "closedin (top_of_set ({0..1} \<times> T)) ({0..1} \<times> S)"
by (simp add: Abstract_Topology.closedin_Times assms)
ultimately have clo0TB:"closedin (top_of_set ({0..1} \<times> T)) B"
by (auto simp: B_def)
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 (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)"
proof (rule continuous_intros)+
show "continuous_on (snd ` ({0} \<times> T)) f"
by (simp add: contf)
qed
ultimately have "continuous_on ({0..1} \<times> S \<union> {0} \<times> T) (\<lambda>x. if snd x \<in> S then h x else (f \<circ> snd) x)"
by (auto intro!: continuous_on_cases_local conth simp: B_def Un_commute [of "{0} \<times> T"])
then have conth': "continuous_on B h'"
by (simp add: h'_def B_def Un_commute [of "{0} \<times> T"])
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 (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
proof
assume ST: "ANR S \<and> ANR T"
have eq: "({0} \<times> T \<inter> {0..1} \<times> S) = {0::real} \<times> S"
using \<open>S \<subseteq> T\<close> by auto
have "ANR B"
unfolding B_def
proof (rule ANR_closed_Un_local)
show "closedin (top_of_set ({0} \<times> T \<union> {0..1} \<times> S)) ({0::real} \<times> T)"
by (metis cloBT B_def)
show "closedin (top_of_set ({0} \<times> T \<union> {0..1} \<times> S)) ({0..1::real} \<times> S)"
by (metis Un_commute cloBS B_def)
qed (simp_all add: ANR_Times convex_imp_ANR ANR_singleton ST eq)
note Vk = that
have *: thesis if "openin (top_of_set ({0..1::real} \<times> T)) V"
"retraction V B r" for V r
proof -
have "continuous_on V (h' \<circ> r)"
using conth' continuous_on_compose retractionE that(2) by blast
moreover have "(h' \<circ> r) ` V \<subseteq> U"
by (metis \<open>h' ` B \<subseteq> U\<close> image_comp retractionE that(2))
ultimately show ?thesis
using Vk [of V "h' \<circ> r"] by (metis comp_apply retraction that)
qed
show thesis
by (meson "*" ANR_imp_neighbourhood_retract \<open>ANR B\<close> clo0TB retract_of_def)
next
assume "ANR U"
with ANR_imp_absolute_neighbourhood_extensor \<open>h' ` B \<subseteq> U\<close> clo0TB conth' that
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 (top_of_set T) S'"
unfolding S'_def using closedin_self opeTV
by (blast intro: closedin_compact_projection)
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 (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"
and "\<And>x. x \<in> T \<Longrightarrow> a x \<in> closed_segment 1 0"
and a1: "\<And>x. x \<in> S \<Longrightarrow> a x = 1"
and a0: "\<And>x. x \<in> S' \<Longrightarrow> a x = 0"
by (rule Urysohn_local [OF cloTS cloTS' \<open>S \<inter> S' = {}\<close>, of 1 0], blast)
then have ain: "\<And>x. x \<in> T \<Longrightarrow> a x \<in> {0..1}"
using closed_segment_eq_real_ivl by auto
have inV: "(u * a t, t) \<in> V" if "t \<in> T" "0 \<le> u" "u \<le> 1" for t u
proof (rule ccontr)
assume "(u * a t, t) \<notin> V"
with ain [OF \<open>t \<in> T\<close>] have "a t = 0"
apply simp
by (metis (no_types, lifting) a0 DiffI S'_def SigmaI atLeastAtMost_iff mem_Collect_eq mult_le_one mult_nonneg_nonneg that)
show False
using B_def \<open>(u * a t, t) \<notin> V\<close> \<open>B \<subseteq> V\<close> \<open>a t = 0\<close> that by auto
qed
show ?thesis
proof
show hom: "homotopic_with_canon (\<lambda>x. True) T U f (\<lambda>x. k (a x, x))"
proof (simp add: homotopic_with, intro exI conjI)
show "continuous_on ({0..1} \<times> T) (k \<circ> (\<lambda>z. (fst z *\<^sub>R (a \<circ> snd) z, snd z)))"
apply (intro continuous_on_compose continuous_intros)
apply (force intro: inV continuous_on_subset [OF contk] continuous_on_subset [OF conta])+
done
show "(k \<circ> (\<lambda>z. (fst z *\<^sub>R (a \<circ> snd) z, snd z))) ` ({0..1} \<times> T) \<subseteq> U"
using inV kim by auto
show "\<forall>x\<in>T. (k \<circ> (\<lambda>z. (fst z *\<^sub>R (a \<circ> snd) z, snd z))) (0, x) = f x"
by (simp add: B_def h'_def keq)
show "\<forall>x\<in>T. (k \<circ> (\<lambda>z. (fst z *\<^sub>R (a \<circ> snd) z, snd z))) (1, x) = k (a x, x)"
by auto
qed
show "continuous_on T (\<lambda>x. k (a x, x))"
using homotopic_with_imp_continuous_maps [OF hom] by auto
show "(\<lambda>x. k (a x, x)) ` T \<subseteq> U"
proof clarify
fix t
assume "t \<in> T"
show "k (a t, t) \<in> U"
by (metis \<open>t \<in> T\<close> image_subset_iff inV kim not_one_le_zero linear mult_cancel_right1)
qed
show "\<And>x. x \<in> S \<Longrightarrow> k (a x, x) = g x"
by (simp add: B_def a1 h'_def keq)
qed
qed
corollary\<^marker>\<open>tag unimportant\<close> nullhomotopic_into_ANR_extension:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "closed S"
and contf: "continuous_on S f"
and "ANR T"
and fim: "f ` S \<subseteq> T"
and "S \<noteq> {}"
shows "(\<exists>c. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c)) \<longleftrightarrow>
(\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> T \<and> (\<forall>x \<in> S. g x = f x))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain c where c: "homotopic_with_canon (\<lambda>x. True) S T (\<lambda>x. c) f"
by (blast intro: homotopic_with_symD)
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"
proof (rule Borsuk_homotopy_extension_homotopic)
show "range (\<lambda>x. c) \<subseteq> T"
using \<open>S \<noteq> {}\<close> c homotopic_with_imp_subset1 by fastforce
qed (use assms c in auto)
then show ?rhs by blast
next
assume ?rhs
then obtain g where "continuous_on UNIV g" "range g \<subseteq> T" "\<And>x. x\<in>S \<Longrightarrow> g x = f x"
by blast
then obtain c where "homotopic_with_canon (\<lambda>h. True) UNIV T g (\<lambda>x. c)"
using nullhomotopic_from_contractible [of UNIV g T] contractible_UNIV by blast
then have "homotopic_with_canon (\<lambda>x. True) S T g (\<lambda>x. c)"
by (simp add: homotopic_from_subtopology)
then show ?lhs
by (force elim: homotopic_with_eq [of _ _ _ g "\<lambda>x. c"] simp: \<open>\<And>x. x \<in> S \<Longrightarrow> g x = f x\<close>)
qed
corollary\<^marker>\<open>tag unimportant\<close> nullhomotopic_into_rel_frontier_extension:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "closed S"
and contf: "continuous_on S f"
and "convex T" "bounded T"
and fim: "f ` S \<subseteq> rel_frontier T"
and "S \<noteq> {}"
shows "(\<exists>c. homotopic_with_canon (\<lambda>x. True) S (rel_frontier T) f (\<lambda>x. c)) \<longleftrightarrow>
(\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> rel_frontier T \<and> (\<forall>x \<in> S. g x = f x))"
by (simp add: nullhomotopic_into_ANR_extension assms ANR_rel_frontier_convex)
corollary\<^marker>\<open>tag unimportant\<close> nullhomotopic_into_sphere_extension:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b :: euclidean_space"
assumes "closed S" and contf: "continuous_on S f"
and "S \<noteq> {}" and fim: "f ` S \<subseteq> sphere a r"
shows "((\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere a r) f (\<lambda>x. c)) \<longleftrightarrow>
(\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> sphere a r \<and> (\<forall>x \<in> S. g x = f x)))"
(is "?lhs = ?rhs")
proof (cases "r = 0")
case True with fim show ?thesis
by (metis ANR_sphere \<open>closed S\<close> \<open>S \<noteq> {}\<close> contf nullhomotopic_into_ANR_extension)
next
case False
then have eq: "sphere a r = rel_frontier (cball a r)" by simp
show ?thesis
using fim nullhomotopic_into_rel_frontier_extension [OF \<open>closed S\<close> contf convex_cball bounded_cball]
by (simp add: \<open>S \<noteq> {}\<close> eq)
qed
proposition\<^marker>\<open>tag unimportant\<close> Borsuk_map_essential_bounded_component:
fixes a :: "'a :: euclidean_space"
assumes "compact S" and "a \<notin> S"
shows "bounded (connected_component_set (- S) a) \<longleftrightarrow>
\<not>(\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere 0 1)
(\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a)) (\<lambda>x. c))"
(is "?lhs = ?rhs")
proof (cases "S = {}")
case True then show ?thesis
by simp
next
case False
have "closed S" "bounded S"
using \<open>compact S\<close> compact_eq_bounded_closed by auto
have s01: "(\<lambda>x. (x - a) /\<^sub>R norm (x - a)) ` S \<subseteq> sphere 0 1"
using \<open>a \<notin> S\<close> by clarsimp (metis dist_eq_0_iff dist_norm mult.commute right_inverse)
have aincc: "a \<in> connected_component_set (- S) a"
by (simp add: \<open>a \<notin> S\<close>)
obtain r where "r>0" and r: "S \<subseteq> ball 0 r"
using bounded_subset_ballD \<open>bounded S\<close> by blast
have "\<not> ?rhs \<longleftrightarrow> \<not> ?lhs"
proof
assume notr: "\<not> ?rhs"
have nog: "\<nexists>g. continuous_on (S \<union> connected_component_set (- S) a) g \<and>
g ` (S \<union> connected_component_set (- S) a) \<subseteq> sphere 0 1 \<and>
(\<forall>x\<in>S. g x = (x - a) /\<^sub>R norm (x - a))"
if "bounded (connected_component_set (- S) a)"
using non_extensible_Borsuk_map [OF \<open>compact S\<close> componentsI _ aincc] \<open>a \<notin> S\<close> that by auto
obtain g where "range g \<subseteq> sphere 0 1" "continuous_on UNIV g"
"\<And>x. x \<in> S \<Longrightarrow> g x = (x - a) /\<^sub>R norm (x - a)"
using notr
by (auto simp: nullhomotopic_into_sphere_extension
[OF \<open>closed S\<close> continuous_on_Borsuk_map [OF \<open>a \<notin> S\<close>] False s01])
with \<open>a \<notin> S\<close> show "\<not> ?lhs"
by (metis UNIV_I continuous_on_subset image_subset_iff nog subsetI)
next
assume "\<not> ?lhs"
then obtain b where b: "b \<in> connected_component_set (- S) a" and "r \<le> norm b"
using bounded_iff linear by blast
then have bnot: "b \<notin> ball 0 r"
by simp
have "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) (\<lambda>x. (x - a) /\<^sub>R norm (x - a))
(\<lambda>x. (x - b) /\<^sub>R norm (x - b))"
proof -
have "path_component (- S) a b"
by (metis (full_types) \<open>closed S\<close> b mem_Collect_eq open_Compl open_path_connected_component)
then show ?thesis
using Borsuk_maps_homotopic_in_path_component by blast
qed
moreover
obtain c where "homotopic_with_canon (\<lambda>x. True) (ball 0 r) (sphere 0 1)
(\<lambda>x. inverse (norm (x - b)) *\<^sub>R (x - b)) (\<lambda>x. c)"
proof (rule nullhomotopic_from_contractible)
show "contractible (ball (0::'a) r)"
by (metis convex_imp_contractible convex_ball)
show "continuous_on (ball 0 r) (\<lambda>x. inverse(norm (x - b)) *\<^sub>R (x - b))"
by (rule continuous_on_Borsuk_map [OF bnot])
show "(\<lambda>x. (x - b) /\<^sub>R norm (x - b)) ` ball 0 r \<subseteq> sphere 0 1"
using bnot Borsuk_map_into_sphere by blast
qed blast
ultimately have "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) (\<lambda>x. (x - a) /\<^sub>R norm (x - a)) (\<lambda>x. c)"
by (meson homotopic_with_subset_left homotopic_with_trans r)
then show "\<not> ?rhs"
by blast
qed
then show ?thesis by blast
qed
lemma homotopic_Borsuk_maps_in_bounded_component:
fixes a :: "'a :: euclidean_space"
assumes "compact S" and "a \<notin> S"and "b \<notin> S"
and boc: "bounded (connected_component_set (- S) a)"
and hom: "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1)
(\<lambda>x. (x - a) /\<^sub>R norm (x - a))
(\<lambda>x. (x - b) /\<^sub>R norm (x - b))"
shows "connected_component (- S) a b"
proof (rule ccontr)
assume notcc: "\<not> connected_component (- S) a b"
let ?T = "S \<union> connected_component_set (- S) a"
have "\<nexists>g. continuous_on (S \<union> connected_component_set (- S) a) g \<and>
g ` (S \<union> connected_component_set (- S) a) \<subseteq> sphere 0 1 \<and>
(\<forall>x\<in>S. g x = (x - a) /\<^sub>R norm (x - a))"
by (simp add: \<open>a \<notin> S\<close> componentsI non_extensible_Borsuk_map [OF \<open>compact S\<close> _ boc])
moreover obtain g where "continuous_on (S \<union> connected_component_set (- S) a) g"
"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 (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)
show "(\<lambda>x. (x - b) /\<^sub>R norm (x - b)) ` ?T \<subseteq> sphere 0 1"
by (simp add: \<open>b \<notin> S\<close> notcc Borsuk_map_into_sphere)
show "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1)
(\<lambda>x. (x - b) /\<^sub>R norm (x - b)) (\<lambda>x. (x - a) /\<^sub>R norm (x - a))"
by (simp add: hom homotopic_with_symD)
qed (auto simp: ANR_sphere intro: that)
ultimately show False by blast
qed
lemma Borsuk_maps_homotopic_in_connected_component_eq:
fixes a :: "'a :: euclidean_space"
assumes S: "compact S" "a \<notin> S" "b \<notin> S" and 2: "2 \<le> DIM('a)"
shows "(homotopic_with_canon (\<lambda>x. True) S (sphere 0 1)
(\<lambda>x. (x - a) /\<^sub>R norm (x - a))
(\<lambda>x. (x - b) /\<^sub>R norm (x - b)) \<longleftrightarrow>
connected_component (- S) a b)"
(is "?lhs = ?rhs")
proof
assume L: ?lhs
show ?rhs
proof (cases "bounded(connected_component_set (- S) a)")
case True
show ?thesis
by (rule homotopic_Borsuk_maps_in_bounded_component [OF S True L])
next
case not_bo_a: False
show ?thesis
proof (cases "bounded(connected_component_set (- S) b)")
case True
show ?thesis
using homotopic_Borsuk_maps_in_bounded_component [OF S]
by (simp add: L True assms connected_component_sym homotopic_Borsuk_maps_in_bounded_component homotopic_with_sym)
next
case False
then show ?thesis
using cobounded_unique_unbounded_component [of "-S" a b] \<open>compact S\<close> not_bo_a
by (auto simp: compact_eq_bounded_closed assms connected_component_eq_eq)
qed
qed
next
assume R: ?rhs
then have "path_component (- S) a b"
using assms(1) compact_eq_bounded_closed open_Compl open_path_connected_component_set by fastforce
then show ?lhs
by (simp add: Borsuk_maps_homotopic_in_path_component)
qed
subsection\<open>More extension theorems\<close>
lemma extension_from_clopen:
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 = {}")
case True
then show ?thesis
by (simp add: null that)
next
case False
then obtain a where "a \<in> U"
by auto
let ?g = "\<lambda>x. if x \<in> T then f x else a"
have Seq: "S = T \<union> (S - T)"
using clo closedin_imp_subset by fastforce
show ?thesis
proof
have "continuous_on (T \<union> (S - T)) ?g"
using Seq clo ope by (intro continuous_on_cases_local) (auto simp: contf)
with Seq show "continuous_on S ?g"
by metis
show "?g ` S \<subseteq> U"
using \<open>a \<in> U\<close> fim by auto
show "\<And>x. x \<in> T \<Longrightarrow> ?g x = f x"
by auto
qed
qed
lemma extension_from_component:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
assumes S: "locally connected S \<or> compact S" and "ANR U"
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 (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
proof
assume "locally connected S"
show ?thesis
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 (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
then obtain V where "open V" and V: "W = S \<inter> V"
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 (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 (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)
show "g ` K \<subseteq> U"
using V \<open>K \<subseteq> V\<close> gim opeK openin_imp_subset by fastforce
qed (use opeK gf \<open>C \<subseteq> K\<close> in auto)
qed
obtain h where "continuous_on S h" "h ` S \<subseteq> U" "\<And>x. x \<in> T \<Longrightarrow> h x = g x"
using extension_from_clopen
by (metis C bot.extremum_uniqueI clo contg gim fim image_is_empty in_components_nonempty ope)
then show ?thesis
by (metis \<open>C \<subseteq> T\<close> gf subset_eq that)
qed
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 (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 (top_of_set (S \<times> T)) (S \<times> T - U)"
using ope by (auto simp: openin_closedin_eq)
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
moreover have "S \<times> (T - ?W) \<subseteq> U"
by auto
ultimately show ?thesis
by (metis (no_types, lifting) Sigma_cong closedin_def that topspace_euclidean_subtopology)
qed
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 (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 (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 (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 (top_of_set T') (\<Union>(F ` T))"
using F by blast
show "T \<subseteq> \<Union>(F ` T)"
using F by blast
show "S \<times> \<Union>(F ` T) \<subseteq> U"
using F by auto
qed
qed
proposition\<^marker>\<open>tag unimportant\<close> homotopic_neighbourhood_extension:
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 (top_of_set S) T"
and "ANR U" and hom: "homotopic_with_canon (\<lambda>x. True) T U f g"
obtains V where "T \<subseteq> V" "openin (top_of_set S) V"
"homotopic_with_canon (\<lambda>x. True) V U f g"
proof -
have "T \<subseteq> S"
using clo closedin_imp_subset by blast
obtain h where conth: "continuous_on ({0..1::real} \<times> T) h"
and him: "h ` ({0..1} \<times> T) \<subseteq> U"
and h0: "\<And>x. h(0, x) = f x" and h1: "\<And>x. h(1, x) = g x"
using hom by (auto simp: homotopic_with_def)
define h' where "h' \<equiv> \<lambda>z. if fst z \<in> {0} then f(snd z)
else if fst z \<in> {1} then g(snd z)
else h z"
let ?S0 = "{0::real} \<times> S" and ?S1 = "{1::real} \<times> S"
have "continuous_on(?S0 \<union> (?S1 \<union> {0..1} \<times> T)) h'"
unfolding h'_def
proof (intro continuous_on_cases_local)
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 (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
show "continuous_on (?S1) (\<lambda>x. g (snd x))"
by (intro continuous_intros continuous_on_compose2 [OF contg]) auto
qed (use h0 h1 conth in auto)
then have "continuous_on ({0,1} \<times> S \<union> ({0..1} \<times> T)) h'"
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 (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 (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 (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_canon (\<lambda>x. True) T' U f g"
proof (simp add: homotopic_with, intro exI conjI)
show "continuous_on ({0..1} \<times> T') k"
using TW continuous_on_subset contk by auto
show "k ` ({0..1} \<times> T') \<subseteq> U"
using TW kim by fastforce
have "T' \<subseteq> S"
by (meson opeT' subsetD openin_imp_subset)
then show "\<forall>x\<in>T'. k (0, x) = f x" "\<forall>x\<in>T'. k (1, x) = g x"
by (auto simp: kh' h'_def)
qed
ultimately show ?thesis
by (blast intro: that)
qed
text\<open> Homotopy on a union of closed-open sets.\<close>
proposition\<^marker>\<open>tag unimportant\<close> homotopic_on_clopen_Union:
fixes \<F> :: "'a::euclidean_space set set"
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_canon (\<lambda>x. True) S T f g"
shows "homotopic_with_canon (\<lambda>x. True) (\<Union>\<F>) T f g"
proof -
obtain \<V> where "\<V> \<subseteq> \<F>" "countable \<V>" and eqU: "\<Union>\<V> = \<Union>\<F>"
using Lindelof_openin assms by blast
show ?thesis
proof (cases "\<V> = {}")
case True
then show ?thesis
by (metis Union_empty eqU homotopic_with_canon_on_empty)
next
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 (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_canon (\<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)"
and him: "\<And>n. h n ` ({0..1} \<times> V n) \<subseteq> T"
and h0: "\<And>n. \<And>x. x \<in> V n \<Longrightarrow> h n (0, x) = f x"
and h1: "\<And>n. \<And>x. x \<in> V n \<Longrightarrow> h n (1, x) = g x"
by (simp add: homotopic_with) metis
have wop: "b \<in> V x \<Longrightarrow> \<exists>k. b \<in> V k \<and> (\<forall>j<k. b \<notin> V j)" for b x
using nat_less_induct [where P = "\<lambda>i. b \<notin> V i"] by meson
obtain \<zeta> where cont: "continuous_on ({0..1} \<times> \<Union>(V ` UNIV)) \<zeta>"
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)
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 (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 (top_of_set (\<Union>(V ` UNIV))) (V i)"
using ope V eqU by auto
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_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
show ?thesis
proof (simp add: homotopic_with eqU [symmetric], intro exI conjI ballI)
show "continuous_on ({0..1} \<times> \<Union>\<V>) \<zeta>"
using V eqU by (blast intro!: continuous_on_subset [OF cont])
show "\<zeta>` ({0..1} \<times> \<Union>\<V>) \<subseteq> T"
proof clarsimp
fix t :: real and y :: "'a" and X :: "'a set"
assume "y \<in> X" "X \<in> \<V>" and t: "0 \<le> t" "t \<le> 1"
then obtain k where "y \<in> V k" and j: "\<forall>j<k. y \<notin> V j"
by (metis image_iff V wop)
with him t show "\<zeta>(t, y) \<in> T"
by (subst eq) force+
qed
fix X y
assume "X \<in> \<V>" "y \<in> X"
then obtain k where "y \<in> V k" and j: "\<forall>j<k. y \<notin> V j"
by (metis image_iff V wop)
then show "\<zeta>(0, y) = f y" and "\<zeta>(1, y) = g y"
by (subst eq [where i=k]; force simp: h0 h1)+
qed
qed
qed
lemma homotopic_on_components_eq:
fixes S :: "'a :: euclidean_space set" and T :: "'b :: euclidean_space set"
assumes S: "locally connected S \<or> compact S" and "ANR T"
shows "homotopic_with_canon (\<lambda>x. True) S T f g \<longleftrightarrow>
(continuous_on S f \<and> f ` S \<subseteq> T \<and> continuous_on S g \<and> g ` S \<subseteq> T) \<and>
(\<forall>C \<in> components S. homotopic_with_canon (\<lambda>x. True) C T f g)"
(is "?lhs \<longleftrightarrow> ?C \<and> ?rhs")
proof -
have "continuous_on S f" "f ` S \<subseteq> T" "continuous_on S g" "g ` S \<subseteq> T" if ?lhs
using homotopic_with_imp_continuous homotopic_with_imp_subset1 homotopic_with_imp_subset2 that by blast+
moreover have "?lhs \<longleftrightarrow> ?rhs"
if contf: "continuous_on S f" and fim: "f ` S \<subseteq> T" and contg: "continuous_on S g" and gim: "g ` S \<subseteq> T"
proof
assume ?lhs
with that show ?rhs
by (simp add: homotopic_with_subset_left in_components_subset)
next
assume R: ?rhs
have "\<exists>U. C \<subseteq> U \<and> closedin (top_of_set S) U \<and>
openin (top_of_set S) U \<and>
homotopic_with_canon (\<lambda>x. True) U T f g" if C: "C \<in> components S" for C
proof -
have "C \<subseteq> S"
by (simp add: in_components_subset that)
show ?thesis
using S
proof
assume "locally connected S"
show ?thesis
proof (intro exI conjI)
show "closedin (top_of_set S) C"
by (simp add: closedin_component that)
show "openin (top_of_set S) C"
by (simp add: \<open>locally connected S\<close> openin_components_locally_connected that)
show "homotopic_with_canon (\<lambda>x. True) C T f g"
by (simp add: R that)
qed auto
next
assume "compact S"
have hom: "homotopic_with_canon (\<lambda>x. True) C T f g"
using R that by blast
obtain U where "C \<subseteq> U" and opeU: "openin (top_of_set S) U"
and hom: "homotopic_with_canon (\<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
then obtain V where "open V" and V: "U = S \<inter> V"
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 (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 (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_canon (\<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
qed (use opeK \<open>C \<subseteq> K\<close> in auto)
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 (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_canon (\<lambda>x. True) (\<phi> C) T f g"
by metis
have Seq: "S = \<Union> (\<phi> ` components S)"
proof
show "S \<subseteq> \<Union> (\<phi> ` components S)"
by (metis Sup_mono Union_components \<phi> imageI)
show "\<Union> (\<phi> ` components S) \<subseteq> S"
using ope\<phi> openin_imp_subset by fastforce
qed
show ?lhs
apply (subst Seq)
using Seq clo\<phi> ope\<phi> hom\<phi> by (intro homotopic_on_clopen_Union) auto
qed
ultimately show ?thesis by blast
qed
lemma cohomotopically_trivial_on_components:
fixes S :: "'a :: euclidean_space set" and T :: "'b :: euclidean_space set"
assumes S: "locally connected S \<or> compact S" and "ANR T"
shows
"(\<forall>f g. continuous_on S f \<longrightarrow> f ` S \<subseteq> T \<longrightarrow> continuous_on S g \<longrightarrow> g ` S \<subseteq> T \<longrightarrow>
homotopic_with_canon (\<lambda>x. True) S T f g)
\<longleftrightarrow>
(\<forall>C\<in>components S.
\<forall>f g. continuous_on C f \<longrightarrow> f ` C \<subseteq> T \<longrightarrow> continuous_on C g \<longrightarrow> g ` C \<subseteq> T \<longrightarrow>
homotopic_with_canon (\<lambda>x. True) C T f g)"
(is "?lhs = ?rhs")
proof
assume L[rule_format]: ?lhs
show ?rhs
proof clarify
fix C f g
assume contf: "continuous_on C f" and fim: "f ` C \<subseteq> T"
and contg: "continuous_on C g" and gim: "g ` C \<subseteq> T" and C: "C \<in> components S"
obtain f' where contf': "continuous_on S f'" and f'im: "f' ` S \<subseteq> T" and f'f: "\<And>x. x \<in> C \<Longrightarrow> f' x = f x"
using extension_from_component [OF S \<open>ANR T\<close> C contf fim] by metis
obtain g' where contg': "continuous_on S g'" and g'im: "g' ` S \<subseteq> T" and g'g: "\<And>x. x \<in> C \<Longrightarrow> g' x = g x"
using extension_from_component [OF S \<open>ANR T\<close> C contg gim] by metis
have "homotopic_with_canon (\<lambda>x. True) C T f' g'"
using L [OF contf' f'im contg' g'im] homotopic_with_subset_left C in_components_subset by fastforce
then show "homotopic_with_canon (\<lambda>x. True) C T f g"
using f'f g'g homotopic_with_eq by force
qed
next
assume R [rule_format]: ?rhs
show ?lhs
proof clarify
fix f g
assume contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
and contg: "continuous_on S g" and gim: "g ` S \<subseteq> T"
moreover have "homotopic_with_canon (\<lambda>x. True) C T f g" if "C \<in> components S" for C
using R [OF that]
by (meson contf contg continuous_on_subset fim gim image_mono in_components_subset order.trans that)
ultimately show "homotopic_with_canon (\<lambda>x. True) S T f g"
by (subst homotopic_on_components_eq [OF S \<open>ANR T\<close>]) auto
qed
qed
subsection\<open>The complement of a set and path-connectedness\<close>
text\<open>Complement in dimension N > 1 of set homeomorphic to any interval in
any dimension is (path-)connected. This naively generalizes the argument
in Ryuji Maehara's paper "The Jordan curve theorem via the Brouwer fixed point theorem",
American Mathematical Monthly 1984.\<close>
lemma unbounded_components_complement_absolute_retract:
fixes S :: "'a::euclidean_space set"
assumes C: "C \<in> components(- S)" and S: "compact S" "AR S"
shows "\<not> bounded C"
proof -
obtain y where y: "C = connected_component_set (- S) y" and "y \<notin> S"
using C by (auto simp: components_def)
have "open(- S)"
using S by (simp add: closed_open compact_eq_bounded_closed)
have "S retract_of UNIV"
using S compact_AR by blast
then obtain r where contr: "continuous_on UNIV r" and ontor: "range r \<subseteq> S"
and r: "\<And>x. x \<in> S \<Longrightarrow> r x = x"
by (auto simp: retract_of_def retraction_def)
show ?thesis
proof
assume "bounded C"
have "connected_component_set (- S) y \<subseteq> S"
proof (rule frontier_subset_retraction)
show "bounded (connected_component_set (- S) y)"
using \<open>bounded C\<close> y by blast
show "frontier (connected_component_set (- S) y) \<subseteq> S"
using C \<open>compact S\<close> compact_eq_bounded_closed frontier_of_components_closed_complement y by blast
show "continuous_on (closure (connected_component_set (- S) y)) r"
by (blast intro: continuous_on_subset [OF contr])
qed (use ontor r in auto)
with \<open>y \<notin> S\<close> show False by force
qed
qed
lemma connected_complement_absolute_retract:
fixes S :: "'a::euclidean_space set"
assumes S: "compact S" "AR S" and 2: "2 \<le> DIM('a)"
shows "connected(- S)"
proof -
have "S retract_of UNIV"
using S compact_AR by blast
show ?thesis
proof (clarsimp simp: connected_iff_connected_component_eq)
have "\<not> bounded (connected_component_set (- S) x)" if "x \<notin> S" for x
by (meson Compl_iff assms componentsI that unbounded_components_complement_absolute_retract)
then show "connected_component_set (- S) x = connected_component_set (- S) y"
if "x \<notin> S" "y \<notin> S" for x y
using cobounded_unique_unbounded_component [OF _ 2]
by (metis \<open>compact S\<close> compact_imp_bounded double_compl that)
qed
qed
lemma path_connected_complement_absolute_retract:
fixes S :: "'a::euclidean_space set"
assumes "compact S" "AR S" "2 \<le> DIM('a)"
shows "path_connected(- S)"
using connected_complement_absolute_retract [OF assms]
using \<open>compact S\<close> compact_eq_bounded_closed connected_open_path_connected by blast
theorem connected_complement_homeomorphic_convex_compact:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes hom: "S homeomorphic T" and T: "convex T" "compact T" and 2: "2 \<le> DIM('a)"
shows "connected(- S)"
proof (cases "S = {}")
case True
then show ?thesis
by (simp add: connected_UNIV)
next
case False
show ?thesis
proof (rule connected_complement_absolute_retract)
show "compact S"
using \<open>compact T\<close> hom homeomorphic_compactness by auto
show "AR S"
by (meson AR_ANR False \<open>convex T\<close> convex_imp_ANR convex_imp_contractible hom homeomorphic_ANR_iff_ANR homeomorphic_contractible_eq)
qed (rule 2)
qed
corollary path_connected_complement_homeomorphic_convex_compact:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes hom: "S homeomorphic T" "convex T" "compact T" "2 \<le> DIM('a)"
shows "path_connected(- S)"
using connected_complement_homeomorphic_convex_compact [OF assms]
using \<open>compact T\<close> compact_eq_bounded_closed connected_open_path_connected hom homeomorphic_compactness by blast
lemma path_connected_complement_homeomorphic_interval:
fixes S :: "'a::euclidean_space set"
assumes "S homeomorphic cbox a b" "2 \<le> DIM('a)"
shows "path_connected(-S)"
using assms compact_cbox convex_box(1) path_connected_complement_homeomorphic_convex_compact by blast
lemma connected_complement_homeomorphic_interval:
fixes S :: "'a::euclidean_space set"
assumes "S homeomorphic cbox a b" "2 \<le> DIM('a)"
shows "connected(-S)"
using assms path_connected_complement_homeomorphic_interval path_connected_imp_connected by blast
end