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1 section \<open>Absolute Retracts, Absolute Neighbourhood Retracts and Euclidean Neighbourhood Retracts\<close> |
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2 |
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3 theory Retracts |
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4 imports Brouwer_Fixpoint Ordered_Euclidean_Space |
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5 begin |
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6 |
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7 |
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8 text \<open>Absolute retracts (AR), absolute neighbourhood retracts (ANR) and also Euclidean neighbourhood |
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9 retracts (ENR). We define AR and ANR by specializing the standard definitions for a set to embedding |
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10 in spaces of higher dimension. |
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11 |
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12 John Harrison writes: "This turns out to be sufficient (since any set in \<open>\<real>\<^sup>n\<close> can be |
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13 embedded as a closed subset of a convex subset of \<open>\<real>\<^sup>n\<^sup>+\<^sup>1\<close>) to derive the usual |
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14 definitions, but we need to split them into two implications because of the lack of type |
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15 quantifiers. Then ENR turns out to be equivalent to ANR plus local compactness."\<close> |
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16 |
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17 definition\<^marker>\<open>tag important\<close> AR :: "'a::topological_space set \<Rightarrow> bool" where |
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18 "AR S \<equiv> \<forall>U. \<forall>S'::('a * real) set. |
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19 S homeomorphic S' \<and> closedin (top_of_set U) S' \<longrightarrow> S' retract_of U" |
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20 |
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21 definition\<^marker>\<open>tag important\<close> ANR :: "'a::topological_space set \<Rightarrow> bool" where |
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22 "ANR S \<equiv> \<forall>U. \<forall>S'::('a * real) set. |
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23 S homeomorphic S' \<and> closedin (top_of_set U) S' |
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24 \<longrightarrow> (\<exists>T. openin (top_of_set U) T \<and> S' retract_of T)" |
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25 |
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26 definition\<^marker>\<open>tag important\<close> ENR :: "'a::topological_space set \<Rightarrow> bool" where |
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27 "ENR S \<equiv> \<exists>U. open U \<and> S retract_of U" |
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28 |
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29 text \<open>First, show that we do indeed get the "usual" properties of ARs and ANRs.\<close> |
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30 |
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31 lemma AR_imp_absolute_extensor: |
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32 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
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33 assumes "AR S" and contf: "continuous_on T f" and "f ` T \<subseteq> S" |
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34 and cloUT: "closedin (top_of_set U) T" |
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35 obtains g where "continuous_on U g" "g ` U \<subseteq> S" "\<And>x. x \<in> T \<Longrightarrow> g x = f x" |
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36 proof - |
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37 have "aff_dim S < int (DIM('b \<times> real))" |
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38 using aff_dim_le_DIM [of S] by simp |
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39 then obtain C and S' :: "('b * real) set" |
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40 where C: "convex C" "C \<noteq> {}" |
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41 and cloCS: "closedin (top_of_set C) S'" |
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42 and hom: "S homeomorphic S'" |
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43 by (metis that homeomorphic_closedin_convex) |
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44 then have "S' retract_of C" |
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45 using \<open>AR S\<close> by (simp add: AR_def) |
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46 then obtain r where "S' \<subseteq> C" and contr: "continuous_on C r" |
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47 and "r ` C \<subseteq> S'" and rid: "\<And>x. x\<in>S' \<Longrightarrow> r x = x" |
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48 by (auto simp: retraction_def retract_of_def) |
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49 obtain g h where "homeomorphism S S' g h" |
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50 using hom by (force simp: homeomorphic_def) |
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51 then have "continuous_on (f ` T) g" |
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52 by (meson \<open>f ` T \<subseteq> S\<close> continuous_on_subset homeomorphism_def) |
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53 then have contgf: "continuous_on T (g \<circ> f)" |
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54 by (metis continuous_on_compose contf) |
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55 have gfTC: "(g \<circ> f) ` T \<subseteq> C" |
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56 proof - |
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57 have "g ` S = S'" |
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58 by (metis (no_types) \<open>homeomorphism S S' g h\<close> homeomorphism_def) |
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59 with \<open>S' \<subseteq> C\<close> \<open>f ` T \<subseteq> S\<close> show ?thesis by force |
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60 qed |
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61 obtain f' where f': "continuous_on U f'" "f' ` U \<subseteq> C" |
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62 "\<And>x. x \<in> T \<Longrightarrow> f' x = (g \<circ> f) x" |
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63 by (metis Dugundji [OF C cloUT contgf gfTC]) |
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64 show ?thesis |
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65 proof (rule_tac g = "h \<circ> r \<circ> f'" in that) |
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66 show "continuous_on U (h \<circ> r \<circ> f')" |
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67 apply (intro continuous_on_compose f') |
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68 using continuous_on_subset contr f' apply blast |
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69 by (meson \<open>homeomorphism S S' g h\<close> \<open>r ` C \<subseteq> S'\<close> continuous_on_subset \<open>f' ` U \<subseteq> C\<close> homeomorphism_def image_mono) |
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70 show "(h \<circ> r \<circ> f') ` U \<subseteq> S" |
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71 using \<open>homeomorphism S S' g h\<close> \<open>r ` C \<subseteq> S'\<close> \<open>f' ` U \<subseteq> C\<close> |
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72 by (fastforce simp: homeomorphism_def) |
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73 show "\<And>x. x \<in> T \<Longrightarrow> (h \<circ> r \<circ> f') x = f x" |
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74 using \<open>homeomorphism S S' g h\<close> \<open>f ` T \<subseteq> S\<close> f' |
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75 by (auto simp: rid homeomorphism_def) |
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76 qed |
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77 qed |
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78 |
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79 lemma AR_imp_absolute_retract: |
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80 fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set" |
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81 assumes "AR S" "S homeomorphic S'" |
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82 and clo: "closedin (top_of_set U) S'" |
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83 shows "S' retract_of U" |
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84 proof - |
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85 obtain g h where hom: "homeomorphism S S' g h" |
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86 using assms by (force simp: homeomorphic_def) |
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87 have h: "continuous_on S' h" " h ` S' \<subseteq> S" |
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88 using hom homeomorphism_def apply blast |
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89 apply (metis hom equalityE homeomorphism_def) |
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90 done |
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91 obtain h' where h': "continuous_on U h'" "h' ` U \<subseteq> S" |
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92 and h'h: "\<And>x. x \<in> S' \<Longrightarrow> h' x = h x" |
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93 by (blast intro: AR_imp_absolute_extensor [OF \<open>AR S\<close> h clo]) |
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94 have [simp]: "S' \<subseteq> U" using clo closedin_limpt by blast |
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95 show ?thesis |
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96 proof (simp add: retraction_def retract_of_def, intro exI conjI) |
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97 show "continuous_on U (g \<circ> h')" |
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98 apply (intro continuous_on_compose h') |
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99 apply (meson hom continuous_on_subset h' homeomorphism_cont1) |
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100 done |
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101 show "(g \<circ> h') ` U \<subseteq> S'" |
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102 using h' by clarsimp (metis hom subsetD homeomorphism_def imageI) |
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103 show "\<forall>x\<in>S'. (g \<circ> h') x = x" |
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104 by clarsimp (metis h'h hom homeomorphism_def) |
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105 qed |
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106 qed |
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107 |
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108 lemma AR_imp_absolute_retract_UNIV: |
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109 fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set" |
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110 assumes "AR S" and hom: "S homeomorphic S'" |
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111 and clo: "closed S'" |
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112 shows "S' retract_of UNIV" |
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113 apply (rule AR_imp_absolute_retract [OF \<open>AR S\<close> hom]) |
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114 using clo closed_closedin by auto |
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115 |
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116 lemma absolute_extensor_imp_AR: |
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117 fixes S :: "'a::euclidean_space set" |
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118 assumes "\<And>f :: 'a * real \<Rightarrow> 'a. |
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119 \<And>U T. \<lbrakk>continuous_on T f; f ` T \<subseteq> S; |
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120 closedin (top_of_set U) T\<rbrakk> |
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121 \<Longrightarrow> \<exists>g. continuous_on U g \<and> g ` U \<subseteq> S \<and> (\<forall>x \<in> T. g x = f x)" |
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122 shows "AR S" |
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123 proof (clarsimp simp: AR_def) |
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124 fix U and T :: "('a * real) set" |
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125 assume "S homeomorphic T" and clo: "closedin (top_of_set U) T" |
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126 then obtain g h where hom: "homeomorphism S T g h" |
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127 by (force simp: homeomorphic_def) |
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128 have h: "continuous_on T h" " h ` T \<subseteq> S" |
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129 using hom homeomorphism_def apply blast |
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130 apply (metis hom equalityE homeomorphism_def) |
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131 done |
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132 obtain h' where h': "continuous_on U h'" "h' ` U \<subseteq> S" |
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133 and h'h: "\<forall>x\<in>T. h' x = h x" |
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134 using assms [OF h clo] by blast |
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135 have [simp]: "T \<subseteq> U" |
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136 using clo closedin_imp_subset by auto |
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137 show "T retract_of U" |
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138 proof (simp add: retraction_def retract_of_def, intro exI conjI) |
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139 show "continuous_on U (g \<circ> h')" |
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140 apply (intro continuous_on_compose h') |
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141 apply (meson hom continuous_on_subset h' homeomorphism_cont1) |
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142 done |
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143 show "(g \<circ> h') ` U \<subseteq> T" |
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144 using h' by clarsimp (metis hom subsetD homeomorphism_def imageI) |
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145 show "\<forall>x\<in>T. (g \<circ> h') x = x" |
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146 by clarsimp (metis h'h hom homeomorphism_def) |
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147 qed |
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148 qed |
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149 |
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150 lemma AR_eq_absolute_extensor: |
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151 fixes S :: "'a::euclidean_space set" |
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152 shows "AR S \<longleftrightarrow> |
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153 (\<forall>f :: 'a * real \<Rightarrow> 'a. |
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154 \<forall>U T. continuous_on T f \<longrightarrow> f ` T \<subseteq> S \<longrightarrow> |
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155 closedin (top_of_set U) T \<longrightarrow> |
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156 (\<exists>g. continuous_on U g \<and> g ` U \<subseteq> S \<and> (\<forall>x \<in> T. g x = f x)))" |
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157 apply (rule iffI) |
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158 apply (metis AR_imp_absolute_extensor) |
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159 apply (simp add: absolute_extensor_imp_AR) |
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160 done |
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161 |
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162 lemma AR_imp_retract: |
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163 fixes S :: "'a::euclidean_space set" |
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164 assumes "AR S \<and> closedin (top_of_set U) S" |
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165 shows "S retract_of U" |
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166 using AR_imp_absolute_retract assms homeomorphic_refl by blast |
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167 |
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168 lemma AR_homeomorphic_AR: |
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169 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
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170 assumes "AR T" "S homeomorphic T" |
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171 shows "AR S" |
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172 unfolding AR_def |
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173 by (metis assms AR_imp_absolute_retract homeomorphic_trans [of _ S] homeomorphic_sym) |
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174 |
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175 lemma homeomorphic_AR_iff_AR: |
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176 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
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177 shows "S homeomorphic T \<Longrightarrow> AR S \<longleftrightarrow> AR T" |
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178 by (metis AR_homeomorphic_AR homeomorphic_sym) |
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179 |
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180 |
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181 lemma ANR_imp_absolute_neighbourhood_extensor: |
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182 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
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183 assumes "ANR S" and contf: "continuous_on T f" and "f ` T \<subseteq> S" |
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184 and cloUT: "closedin (top_of_set U) T" |
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185 obtains V g where "T \<subseteq> V" "openin (top_of_set U) V" |
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186 "continuous_on V g" |
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187 "g ` V \<subseteq> S" "\<And>x. x \<in> T \<Longrightarrow> g x = f x" |
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188 proof - |
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189 have "aff_dim S < int (DIM('b \<times> real))" |
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190 using aff_dim_le_DIM [of S] by simp |
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191 then obtain C and S' :: "('b * real) set" |
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192 where C: "convex C" "C \<noteq> {}" |
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193 and cloCS: "closedin (top_of_set C) S'" |
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194 and hom: "S homeomorphic S'" |
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195 by (metis that homeomorphic_closedin_convex) |
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196 then obtain D where opD: "openin (top_of_set C) D" and "S' retract_of D" |
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197 using \<open>ANR S\<close> by (auto simp: ANR_def) |
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198 then obtain r where "S' \<subseteq> D" and contr: "continuous_on D r" |
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199 and "r ` D \<subseteq> S'" and rid: "\<And>x. x \<in> S' \<Longrightarrow> r x = x" |
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200 by (auto simp: retraction_def retract_of_def) |
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201 obtain g h where homgh: "homeomorphism S S' g h" |
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202 using hom by (force simp: homeomorphic_def) |
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203 have "continuous_on (f ` T) g" |
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204 by (meson \<open>f ` T \<subseteq> S\<close> continuous_on_subset homeomorphism_def homgh) |
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205 then have contgf: "continuous_on T (g \<circ> f)" |
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206 by (intro continuous_on_compose contf) |
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207 have gfTC: "(g \<circ> f) ` T \<subseteq> C" |
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208 proof - |
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209 have "g ` S = S'" |
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210 by (metis (no_types) homeomorphism_def homgh) |
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211 then show ?thesis |
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212 by (metis (no_types) assms(3) cloCS closedin_def image_comp image_mono order.trans topspace_euclidean_subtopology) |
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213 qed |
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214 obtain f' where contf': "continuous_on U f'" |
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215 and "f' ` U \<subseteq> C" |
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216 and eq: "\<And>x. x \<in> T \<Longrightarrow> f' x = (g \<circ> f) x" |
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217 by (metis Dugundji [OF C cloUT contgf gfTC]) |
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218 show ?thesis |
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219 proof (rule_tac V = "U \<inter> f' -` D" and g = "h \<circ> r \<circ> f'" in that) |
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220 show "T \<subseteq> U \<inter> f' -` D" |
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221 using cloUT closedin_imp_subset \<open>S' \<subseteq> D\<close> \<open>f ` T \<subseteq> S\<close> eq homeomorphism_image1 homgh |
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222 by fastforce |
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223 show ope: "openin (top_of_set U) (U \<inter> f' -` D)" |
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224 using \<open>f' ` U \<subseteq> C\<close> by (auto simp: opD contf' continuous_openin_preimage) |
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225 have conth: "continuous_on (r ` f' ` (U \<inter> f' -` D)) h" |
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226 apply (rule continuous_on_subset [of S']) |
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227 using homeomorphism_def homgh apply blast |
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228 using \<open>r ` D \<subseteq> S'\<close> by blast |
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229 show "continuous_on (U \<inter> f' -` D) (h \<circ> r \<circ> f')" |
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230 apply (intro continuous_on_compose conth |
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231 continuous_on_subset [OF contr] continuous_on_subset [OF contf'], auto) |
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232 done |
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233 show "(h \<circ> r \<circ> f') ` (U \<inter> f' -` D) \<subseteq> S" |
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234 using \<open>homeomorphism S S' g h\<close> \<open>f' ` U \<subseteq> C\<close> \<open>r ` D \<subseteq> S'\<close> |
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235 by (auto simp: homeomorphism_def) |
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236 show "\<And>x. x \<in> T \<Longrightarrow> (h \<circ> r \<circ> f') x = f x" |
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237 using \<open>homeomorphism S S' g h\<close> \<open>f ` T \<subseteq> S\<close> eq |
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238 by (auto simp: rid homeomorphism_def) |
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239 qed |
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240 qed |
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241 |
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242 |
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243 corollary ANR_imp_absolute_neighbourhood_retract: |
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244 fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set" |
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245 assumes "ANR S" "S homeomorphic S'" |
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246 and clo: "closedin (top_of_set U) S'" |
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247 obtains V where "openin (top_of_set U) V" "S' retract_of V" |
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248 proof - |
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249 obtain g h where hom: "homeomorphism S S' g h" |
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250 using assms by (force simp: homeomorphic_def) |
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251 have h: "continuous_on S' h" " h ` S' \<subseteq> S" |
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252 using hom homeomorphism_def apply blast |
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253 apply (metis hom equalityE homeomorphism_def) |
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254 done |
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255 from ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> h clo] |
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256 obtain V h' where "S' \<subseteq> V" and opUV: "openin (top_of_set U) V" |
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257 and h': "continuous_on V h'" "h' ` V \<subseteq> S" |
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258 and h'h:"\<And>x. x \<in> S' \<Longrightarrow> h' x = h x" |
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259 by (blast intro: ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> h clo]) |
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260 have "S' retract_of V" |
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261 proof (simp add: retraction_def retract_of_def, intro exI conjI \<open>S' \<subseteq> V\<close>) |
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262 show "continuous_on V (g \<circ> h')" |
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263 apply (intro continuous_on_compose h') |
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264 apply (meson hom continuous_on_subset h' homeomorphism_cont1) |
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265 done |
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266 show "(g \<circ> h') ` V \<subseteq> S'" |
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267 using h' by clarsimp (metis hom subsetD homeomorphism_def imageI) |
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268 show "\<forall>x\<in>S'. (g \<circ> h') x = x" |
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269 by clarsimp (metis h'h hom homeomorphism_def) |
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270 qed |
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271 then show ?thesis |
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272 by (rule that [OF opUV]) |
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273 qed |
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274 |
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275 corollary ANR_imp_absolute_neighbourhood_retract_UNIV: |
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276 fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set" |
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277 assumes "ANR S" and hom: "S homeomorphic S'" and clo: "closed S'" |
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278 obtains V where "open V" "S' retract_of V" |
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279 using ANR_imp_absolute_neighbourhood_retract [OF \<open>ANR S\<close> hom] |
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280 by (metis clo closed_closedin open_openin subtopology_UNIV) |
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281 |
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282 corollary neighbourhood_extension_into_ANR: |
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283 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
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284 assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T" and "ANR T" "closed S" |
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285 obtains V g where "S \<subseteq> V" "open V" "continuous_on V g" |
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286 "g ` V \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
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287 using ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> contf fim] |
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288 by (metis \<open>closed S\<close> closed_closedin open_openin subtopology_UNIV) |
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289 |
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290 lemma absolute_neighbourhood_extensor_imp_ANR: |
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291 fixes S :: "'a::euclidean_space set" |
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292 assumes "\<And>f :: 'a * real \<Rightarrow> 'a. |
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293 \<And>U T. \<lbrakk>continuous_on T f; f ` T \<subseteq> S; |
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294 closedin (top_of_set U) T\<rbrakk> |
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295 \<Longrightarrow> \<exists>V g. T \<subseteq> V \<and> openin (top_of_set U) V \<and> |
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296 continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x \<in> T. g x = f x)" |
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297 shows "ANR S" |
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298 proof (clarsimp simp: ANR_def) |
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299 fix U and T :: "('a * real) set" |
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300 assume "S homeomorphic T" and clo: "closedin (top_of_set U) T" |
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301 then obtain g h where hom: "homeomorphism S T g h" |
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302 by (force simp: homeomorphic_def) |
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303 have h: "continuous_on T h" " h ` T \<subseteq> S" |
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304 using hom homeomorphism_def apply blast |
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305 apply (metis hom equalityE homeomorphism_def) |
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306 done |
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307 obtain V h' where "T \<subseteq> V" and opV: "openin (top_of_set U) V" |
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308 and h': "continuous_on V h'" "h' ` V \<subseteq> S" |
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309 and h'h: "\<forall>x\<in>T. h' x = h x" |
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310 using assms [OF h clo] by blast |
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311 have [simp]: "T \<subseteq> U" |
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312 using clo closedin_imp_subset by auto |
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313 have "T retract_of V" |
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314 proof (simp add: retraction_def retract_of_def, intro exI conjI \<open>T \<subseteq> V\<close>) |
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315 show "continuous_on V (g \<circ> h')" |
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316 apply (intro continuous_on_compose h') |
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317 apply (meson hom continuous_on_subset h' homeomorphism_cont1) |
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318 done |
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319 show "(g \<circ> h') ` V \<subseteq> T" |
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320 using h' by clarsimp (metis hom subsetD homeomorphism_def imageI) |
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321 show "\<forall>x\<in>T. (g \<circ> h') x = x" |
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322 by clarsimp (metis h'h hom homeomorphism_def) |
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323 qed |
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324 then show "\<exists>V. openin (top_of_set U) V \<and> T retract_of V" |
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325 using opV by blast |
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326 qed |
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327 |
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328 lemma ANR_eq_absolute_neighbourhood_extensor: |
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329 fixes S :: "'a::euclidean_space set" |
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330 shows "ANR S \<longleftrightarrow> |
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331 (\<forall>f :: 'a * real \<Rightarrow> 'a. |
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332 \<forall>U T. continuous_on T f \<longrightarrow> f ` T \<subseteq> S \<longrightarrow> |
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333 closedin (top_of_set U) T \<longrightarrow> |
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334 (\<exists>V g. T \<subseteq> V \<and> openin (top_of_set U) V \<and> |
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335 continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x \<in> T. g x = f x)))" |
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336 apply (rule iffI) |
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337 apply (metis ANR_imp_absolute_neighbourhood_extensor) |
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338 apply (simp add: absolute_neighbourhood_extensor_imp_ANR) |
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339 done |
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340 |
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341 lemma ANR_imp_neighbourhood_retract: |
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342 fixes S :: "'a::euclidean_space set" |
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343 assumes "ANR S" "closedin (top_of_set U) S" |
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344 obtains V where "openin (top_of_set U) V" "S retract_of V" |
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345 using ANR_imp_absolute_neighbourhood_retract assms homeomorphic_refl by blast |
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346 |
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347 lemma ANR_imp_absolute_closed_neighbourhood_retract: |
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348 fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set" |
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349 assumes "ANR S" "S homeomorphic S'" and US': "closedin (top_of_set U) S'" |
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350 obtains V W |
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351 where "openin (top_of_set U) V" |
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352 "closedin (top_of_set U) W" |
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353 "S' \<subseteq> V" "V \<subseteq> W" "S' retract_of W" |
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354 proof - |
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355 obtain Z where "openin (top_of_set U) Z" and S'Z: "S' retract_of Z" |
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356 by (blast intro: assms ANR_imp_absolute_neighbourhood_retract) |
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357 then have UUZ: "closedin (top_of_set U) (U - Z)" |
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358 by auto |
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359 have "S' \<inter> (U - Z) = {}" |
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360 using \<open>S' retract_of Z\<close> closedin_retract closedin_subtopology by fastforce |
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361 then obtain V W |
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362 where "openin (top_of_set U) V" |
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363 and "openin (top_of_set U) W" |
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364 and "S' \<subseteq> V" "U - Z \<subseteq> W" "V \<inter> W = {}" |
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365 using separation_normal_local [OF US' UUZ] by auto |
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366 moreover have "S' retract_of U - W" |
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367 apply (rule retract_of_subset [OF S'Z]) |
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368 using US' \<open>S' \<subseteq> V\<close> \<open>V \<inter> W = {}\<close> closedin_subset apply fastforce |
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369 using Diff_subset_conv \<open>U - Z \<subseteq> W\<close> by blast |
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370 ultimately show ?thesis |
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371 apply (rule_tac V=V and W = "U-W" in that) |
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372 using openin_imp_subset apply force+ |
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373 done |
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374 qed |
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375 |
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376 lemma ANR_imp_closed_neighbourhood_retract: |
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377 fixes S :: "'a::euclidean_space set" |
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378 assumes "ANR S" "closedin (top_of_set U) S" |
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379 obtains V W where "openin (top_of_set U) V" |
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380 "closedin (top_of_set U) W" |
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381 "S \<subseteq> V" "V \<subseteq> W" "S retract_of W" |
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382 by (meson ANR_imp_absolute_closed_neighbourhood_retract assms homeomorphic_refl) |
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383 |
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384 lemma ANR_homeomorphic_ANR: |
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385 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
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386 assumes "ANR T" "S homeomorphic T" |
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387 shows "ANR S" |
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388 unfolding ANR_def |
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389 by (metis assms ANR_imp_absolute_neighbourhood_retract homeomorphic_trans [of _ S] homeomorphic_sym) |
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390 |
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391 lemma homeomorphic_ANR_iff_ANR: |
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392 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
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393 shows "S homeomorphic T \<Longrightarrow> ANR S \<longleftrightarrow> ANR T" |
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394 by (metis ANR_homeomorphic_ANR homeomorphic_sym) |
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395 |
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396 subsection \<open>Analogous properties of ENRs\<close> |
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397 |
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398 lemma ENR_imp_absolute_neighbourhood_retract: |
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399 fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set" |
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400 assumes "ENR S" and hom: "S homeomorphic S'" |
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401 and "S' \<subseteq> U" |
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402 obtains V where "openin (top_of_set U) V" "S' retract_of V" |
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403 proof - |
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404 obtain X where "open X" "S retract_of X" |
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405 using \<open>ENR S\<close> by (auto simp: ENR_def) |
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406 then obtain r where "retraction X S r" |
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407 by (auto simp: retract_of_def) |
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408 have "locally compact S'" |
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409 using retract_of_locally_compact open_imp_locally_compact |
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410 homeomorphic_local_compactness \<open>S retract_of X\<close> \<open>open X\<close> hom by blast |
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411 then obtain W where UW: "openin (top_of_set U) W" |
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412 and WS': "closedin (top_of_set W) S'" |
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413 apply (rule locally_compact_closedin_open) |
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414 apply (rename_tac W) |
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415 apply (rule_tac W = "U \<inter> W" in that, blast) |
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416 by (simp add: \<open>S' \<subseteq> U\<close> closedin_limpt) |
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417 obtain f g where hom: "homeomorphism S S' f g" |
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418 using assms by (force simp: homeomorphic_def) |
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419 have contg: "continuous_on S' g" |
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420 using hom homeomorphism_def by blast |
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421 moreover have "g ` S' \<subseteq> S" by (metis hom equalityE homeomorphism_def) |
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422 ultimately obtain h where conth: "continuous_on W h" and hg: "\<And>x. x \<in> S' \<Longrightarrow> h x = g x" |
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423 using Tietze_unbounded [of S' g W] WS' by blast |
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424 have "W \<subseteq> U" using UW openin_open by auto |
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425 have "S' \<subseteq> W" using WS' closedin_closed by auto |
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426 have him: "\<And>x. x \<in> S' \<Longrightarrow> h x \<in> X" |
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427 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) |
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428 have "S' retract_of (W \<inter> h -` X)" |
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429 proof (simp add: retraction_def retract_of_def, intro exI conjI) |
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430 show "S' \<subseteq> W" "S' \<subseteq> h -` X" |
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431 using him WS' closedin_imp_subset by blast+ |
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432 show "continuous_on (W \<inter> h -` X) (f \<circ> r \<circ> h)" |
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433 proof (intro continuous_on_compose) |
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434 show "continuous_on (W \<inter> h -` X) h" |
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435 by (meson conth continuous_on_subset inf_le1) |
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436 show "continuous_on (h ` (W \<inter> h -` X)) r" |
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437 proof - |
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438 have "h ` (W \<inter> h -` X) \<subseteq> X" |
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439 by blast |
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440 then show "continuous_on (h ` (W \<inter> h -` X)) r" |
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441 by (meson \<open>retraction X S r\<close> continuous_on_subset retraction) |
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442 qed |
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443 show "continuous_on (r ` h ` (W \<inter> h -` X)) f" |
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444 apply (rule continuous_on_subset [of S]) |
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445 using hom homeomorphism_def apply blast |
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446 apply clarify |
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447 apply (meson \<open>retraction X S r\<close> subsetD imageI retraction_def) |
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448 done |
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449 qed |
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450 show "(f \<circ> r \<circ> h) ` (W \<inter> h -` X) \<subseteq> S'" |
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451 using \<open>retraction X S r\<close> hom |
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452 by (auto simp: retraction_def homeomorphism_def) |
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453 show "\<forall>x\<in>S'. (f \<circ> r \<circ> h) x = x" |
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454 using \<open>retraction X S r\<close> hom by (auto simp: retraction_def homeomorphism_def hg) |
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455 qed |
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456 then show ?thesis |
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457 apply (rule_tac V = "W \<inter> h -` X" in that) |
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458 apply (rule openin_trans [OF _ UW]) |
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459 using \<open>continuous_on W h\<close> \<open>open X\<close> continuous_openin_preimage_eq apply blast+ |
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460 done |
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461 qed |
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462 |
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463 corollary ENR_imp_absolute_neighbourhood_retract_UNIV: |
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464 fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set" |
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465 assumes "ENR S" "S homeomorphic S'" |
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466 obtains T' where "open T'" "S' retract_of T'" |
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467 by (metis ENR_imp_absolute_neighbourhood_retract UNIV_I assms(1) assms(2) open_openin subsetI subtopology_UNIV) |
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468 |
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469 lemma ENR_homeomorphic_ENR: |
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470 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
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471 assumes "ENR T" "S homeomorphic T" |
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472 shows "ENR S" |
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473 unfolding ENR_def |
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474 by (meson ENR_imp_absolute_neighbourhood_retract_UNIV assms homeomorphic_sym) |
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475 |
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476 lemma homeomorphic_ENR_iff_ENR: |
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477 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
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478 assumes "S homeomorphic T" |
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479 shows "ENR S \<longleftrightarrow> ENR T" |
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480 by (meson ENR_homeomorphic_ENR assms homeomorphic_sym) |
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481 |
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482 lemma ENR_translation: |
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483 fixes S :: "'a::euclidean_space set" |
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484 shows "ENR(image (\<lambda>x. a + x) S) \<longleftrightarrow> ENR S" |
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485 by (meson homeomorphic_sym homeomorphic_translation homeomorphic_ENR_iff_ENR) |
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486 |
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487 lemma ENR_linear_image_eq: |
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488 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
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489 assumes "linear f" "inj f" |
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490 shows "ENR (image f S) \<longleftrightarrow> ENR S" |
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491 apply (rule homeomorphic_ENR_iff_ENR) |
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492 using assms homeomorphic_sym linear_homeomorphic_image by auto |
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493 |
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494 text \<open>Some relations among the concepts. We also relate AR to being a retract of UNIV, which is |
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495 often a more convenient proxy in the closed case.\<close> |
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496 |
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497 lemma AR_imp_ANR: "AR S \<Longrightarrow> ANR S" |
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498 using ANR_def AR_def by fastforce |
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499 |
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500 lemma ENR_imp_ANR: |
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501 fixes S :: "'a::euclidean_space set" |
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502 shows "ENR S \<Longrightarrow> ANR S" |
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503 apply (simp add: ANR_def) |
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504 by (metis ENR_imp_absolute_neighbourhood_retract closedin_imp_subset) |
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505 |
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506 lemma ENR_ANR: |
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507 fixes S :: "'a::euclidean_space set" |
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508 shows "ENR S \<longleftrightarrow> ANR S \<and> locally compact S" |
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509 proof |
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510 assume "ENR S" |
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511 then have "locally compact S" |
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512 using ENR_def open_imp_locally_compact retract_of_locally_compact by auto |
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513 then show "ANR S \<and> locally compact S" |
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514 using ENR_imp_ANR \<open>ENR S\<close> by blast |
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515 next |
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516 assume "ANR S \<and> locally compact S" |
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517 then have "ANR S" "locally compact S" by auto |
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518 then obtain T :: "('a * real) set" where "closed T" "S homeomorphic T" |
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519 using locally_compact_homeomorphic_closed |
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520 by (metis DIM_prod DIM_real Suc_eq_plus1 lessI) |
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521 then show "ENR S" |
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522 using \<open>ANR S\<close> |
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523 apply (simp add: ANR_def) |
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524 apply (drule_tac x=UNIV in spec) |
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525 apply (drule_tac x=T in spec, clarsimp) |
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526 apply (meson ENR_def ENR_homeomorphic_ENR open_openin) |
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527 done |
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528 qed |
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529 |
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530 |
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531 lemma AR_ANR: |
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532 fixes S :: "'a::euclidean_space set" |
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533 shows "AR S \<longleftrightarrow> ANR S \<and> contractible S \<and> S \<noteq> {}" |
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534 (is "?lhs = ?rhs") |
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535 proof |
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536 assume ?lhs |
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537 obtain C and S' :: "('a * real) set" |
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538 where "convex C" "C \<noteq> {}" "closedin (top_of_set C) S'" "S homeomorphic S'" |
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539 apply (rule homeomorphic_closedin_convex [of S, where 'n = "'a * real"]) |
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540 using aff_dim_le_DIM [of S] by auto |
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541 with \<open>AR S\<close> have "contractible S" |
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542 apply (simp add: AR_def) |
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543 apply (drule_tac x=C in spec) |
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544 apply (drule_tac x="S'" in spec, simp) |
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545 using convex_imp_contractible homeomorphic_contractible_eq retract_of_contractible by fastforce |
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546 with \<open>AR S\<close> show ?rhs |
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547 apply (auto simp: AR_imp_ANR) |
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548 apply (force simp: AR_def) |
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549 done |
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550 next |
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551 assume ?rhs |
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552 then obtain a and h:: "real \<times> 'a \<Rightarrow> 'a" |
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553 where conth: "continuous_on ({0..1} \<times> S) h" |
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554 and hS: "h ` ({0..1} \<times> S) \<subseteq> S" |
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555 and [simp]: "\<And>x. h(0, x) = x" |
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556 and [simp]: "\<And>x. h(1, x) = a" |
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557 and "ANR S" "S \<noteq> {}" |
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558 by (auto simp: contractible_def homotopic_with_def) |
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559 then have "a \<in> S" |
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560 by (metis all_not_in_conv atLeastAtMost_iff image_subset_iff mem_Sigma_iff order_refl zero_le_one) |
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561 have "\<exists>g. continuous_on W g \<and> g ` W \<subseteq> S \<and> (\<forall>x\<in>T. g x = f x)" |
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562 if f: "continuous_on T f" "f ` T \<subseteq> S" |
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563 and WT: "closedin (top_of_set W) T" |
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564 for W T and f :: "'a \<times> real \<Rightarrow> 'a" |
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565 proof - |
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566 obtain U g |
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567 where "T \<subseteq> U" and WU: "openin (top_of_set W) U" |
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568 and contg: "continuous_on U g" |
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569 and "g ` U \<subseteq> S" and gf: "\<And>x. x \<in> T \<Longrightarrow> g x = f x" |
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570 using iffD1 [OF ANR_eq_absolute_neighbourhood_extensor \<open>ANR S\<close>, rule_format, OF f WT] |
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571 by auto |
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572 have WWU: "closedin (top_of_set W) (W - U)" |
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573 using WU closedin_diff by fastforce |
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574 moreover have "(W - U) \<inter> T = {}" |
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575 using \<open>T \<subseteq> U\<close> by auto |
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576 ultimately obtain V V' |
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577 where WV': "openin (top_of_set W) V'" |
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578 and WV: "openin (top_of_set W) V" |
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579 and "W - U \<subseteq> V'" "T \<subseteq> V" "V' \<inter> V = {}" |
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580 using separation_normal_local [of W "W-U" T] WT by blast |
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581 then have WVT: "T \<inter> (W - V) = {}" |
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582 by auto |
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583 have WWV: "closedin (top_of_set W) (W - V)" |
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584 using WV closedin_diff by fastforce |
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585 obtain j :: " 'a \<times> real \<Rightarrow> real" |
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586 where contj: "continuous_on W j" |
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587 and j: "\<And>x. x \<in> W \<Longrightarrow> j x \<in> {0..1}" |
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588 and j0: "\<And>x. x \<in> W - V \<Longrightarrow> j x = 1" |
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589 and j1: "\<And>x. x \<in> T \<Longrightarrow> j x = 0" |
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590 by (rule Urysohn_local [OF WT WWV WVT, of 0 "1::real"]) (auto simp: in_segment) |
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591 have Weq: "W = (W - V) \<union> (W - V')" |
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592 using \<open>V' \<inter> V = {}\<close> by force |
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593 show ?thesis |
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594 proof (intro conjI exI) |
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595 have *: "continuous_on (W - V') (\<lambda>x. h (j x, g x))" |
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596 apply (rule continuous_on_compose2 [OF conth continuous_on_Pair]) |
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597 apply (rule continuous_on_subset [OF contj Diff_subset]) |
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598 apply (rule continuous_on_subset [OF contg]) |
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599 apply (metis Diff_subset_conv Un_commute \<open>W - U \<subseteq> V'\<close>) |
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600 using j \<open>g ` U \<subseteq> S\<close> \<open>W - U \<subseteq> V'\<close> apply fastforce |
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601 done |
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602 show "continuous_on W (\<lambda>x. if x \<in> W - V then a else h (j x, g x))" |
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603 apply (subst Weq) |
|
604 apply (rule continuous_on_cases_local) |
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605 apply (simp_all add: Weq [symmetric] WWV continuous_on_const *) |
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606 using WV' closedin_diff apply fastforce |
|
607 apply (auto simp: j0 j1) |
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608 done |
|
609 next |
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610 have "h (j (x, y), g (x, y)) \<in> S" if "(x, y) \<in> W" "(x, y) \<in> V" for x y |
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611 proof - |
|
612 have "j(x, y) \<in> {0..1}" |
|
613 using j that by blast |
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614 moreover have "g(x, y) \<in> S" |
|
615 using \<open>V' \<inter> V = {}\<close> \<open>W - U \<subseteq> V'\<close> \<open>g ` U \<subseteq> S\<close> that by fastforce |
|
616 ultimately show ?thesis |
|
617 using hS by blast |
|
618 qed |
|
619 with \<open>a \<in> S\<close> \<open>g ` U \<subseteq> S\<close> |
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620 show "(\<lambda>x. if x \<in> W - V then a else h (j x, g x)) ` W \<subseteq> S" |
|
621 by auto |
|
622 next |
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623 show "\<forall>x\<in>T. (if x \<in> W - V then a else h (j x, g x)) = f x" |
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624 using \<open>T \<subseteq> V\<close> by (auto simp: j0 j1 gf) |
|
625 qed |
|
626 qed |
|
627 then show ?lhs |
|
628 by (simp add: AR_eq_absolute_extensor) |
|
629 qed |
|
630 |
|
631 |
|
632 lemma ANR_retract_of_ANR: |
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633 fixes S :: "'a::euclidean_space set" |
|
634 assumes "ANR T" "S retract_of T" |
|
635 shows "ANR S" |
|
636 using assms |
|
637 apply (simp add: ANR_eq_absolute_neighbourhood_extensor retract_of_def retraction_def) |
|
638 apply (clarsimp elim!: all_forward) |
|
639 apply (erule impCE, metis subset_trans) |
|
640 apply (clarsimp elim!: ex_forward) |
|
641 apply (rule_tac x="r \<circ> g" in exI) |
|
642 by (metis comp_apply continuous_on_compose continuous_on_subset subsetD imageI image_comp image_mono subset_trans) |
|
643 |
|
644 lemma AR_retract_of_AR: |
|
645 fixes S :: "'a::euclidean_space set" |
|
646 shows "\<lbrakk>AR T; S retract_of T\<rbrakk> \<Longrightarrow> AR S" |
|
647 using ANR_retract_of_ANR AR_ANR retract_of_contractible by fastforce |
|
648 |
|
649 lemma ENR_retract_of_ENR: |
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650 "\<lbrakk>ENR T; S retract_of T\<rbrakk> \<Longrightarrow> ENR S" |
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651 by (meson ENR_def retract_of_trans) |
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652 |
|
653 lemma retract_of_UNIV: |
|
654 fixes S :: "'a::euclidean_space set" |
|
655 shows "S retract_of UNIV \<longleftrightarrow> AR S \<and> closed S" |
|
656 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) |
|
657 |
|
658 lemma compact_AR: |
|
659 fixes S :: "'a::euclidean_space set" |
|
660 shows "compact S \<and> AR S \<longleftrightarrow> compact S \<and> S retract_of UNIV" |
|
661 using compact_imp_closed retract_of_UNIV by blast |
|
662 |
|
663 text \<open>More properties of ARs, ANRs and ENRs\<close> |
|
664 |
|
665 lemma not_AR_empty [simp]: "\<not> AR({})" |
|
666 by (auto simp: AR_def) |
|
667 |
|
668 lemma ENR_empty [simp]: "ENR {}" |
|
669 by (simp add: ENR_def) |
|
670 |
|
671 lemma ANR_empty [simp]: "ANR ({} :: 'a::euclidean_space set)" |
|
672 by (simp add: ENR_imp_ANR) |
|
673 |
|
674 lemma convex_imp_AR: |
|
675 fixes S :: "'a::euclidean_space set" |
|
676 shows "\<lbrakk>convex S; S \<noteq> {}\<rbrakk> \<Longrightarrow> AR S" |
|
677 apply (rule absolute_extensor_imp_AR) |
|
678 apply (rule Dugundji, assumption+) |
|
679 by blast |
|
680 |
|
681 lemma convex_imp_ANR: |
|
682 fixes S :: "'a::euclidean_space set" |
|
683 shows "convex S \<Longrightarrow> ANR S" |
|
684 using ANR_empty AR_imp_ANR convex_imp_AR by blast |
|
685 |
|
686 lemma ENR_convex_closed: |
|
687 fixes S :: "'a::euclidean_space set" |
|
688 shows "\<lbrakk>closed S; convex S\<rbrakk> \<Longrightarrow> ENR S" |
|
689 using ENR_def ENR_empty convex_imp_AR retract_of_UNIV by blast |
|
690 |
|
691 lemma AR_UNIV [simp]: "AR (UNIV :: 'a::euclidean_space set)" |
|
692 using retract_of_UNIV by auto |
|
693 |
|
694 lemma ANR_UNIV [simp]: "ANR (UNIV :: 'a::euclidean_space set)" |
|
695 by (simp add: AR_imp_ANR) |
|
696 |
|
697 lemma ENR_UNIV [simp]:"ENR UNIV" |
|
698 using ENR_def by blast |
|
699 |
|
700 lemma AR_singleton: |
|
701 fixes a :: "'a::euclidean_space" |
|
702 shows "AR {a}" |
|
703 using retract_of_UNIV by blast |
|
704 |
|
705 lemma ANR_singleton: |
|
706 fixes a :: "'a::euclidean_space" |
|
707 shows "ANR {a}" |
|
708 by (simp add: AR_imp_ANR AR_singleton) |
|
709 |
|
710 lemma ENR_singleton: "ENR {a}" |
|
711 using ENR_def by blast |
|
712 |
|
713 text \<open>ARs closed under union\<close> |
|
714 |
|
715 lemma AR_closed_Un_local_aux: |
|
716 fixes U :: "'a::euclidean_space set" |
|
717 assumes "closedin (top_of_set U) S" |
|
718 "closedin (top_of_set U) T" |
|
719 "AR S" "AR T" "AR(S \<inter> T)" |
|
720 shows "(S \<union> T) retract_of U" |
|
721 proof - |
|
722 have "S \<inter> T \<noteq> {}" |
|
723 using assms AR_def by fastforce |
|
724 have "S \<subseteq> U" "T \<subseteq> U" |
|
725 using assms by (auto simp: closedin_imp_subset) |
|
726 define S' where "S' \<equiv> {x \<in> U. setdist {x} S \<le> setdist {x} T}" |
|
727 define T' where "T' \<equiv> {x \<in> U. setdist {x} T \<le> setdist {x} S}" |
|
728 define W where "W \<equiv> {x \<in> U. setdist {x} S = setdist {x} T}" |
|
729 have US': "closedin (top_of_set U) S'" |
|
730 using continuous_closedin_preimage [of U "\<lambda>x. setdist {x} S - setdist {x} T" "{..0}"] |
|
731 by (simp add: S'_def vimage_def Collect_conj_eq continuous_on_diff continuous_on_setdist) |
|
732 have UT': "closedin (top_of_set U) T'" |
|
733 using continuous_closedin_preimage [of U "\<lambda>x. setdist {x} T - setdist {x} S" "{..0}"] |
|
734 by (simp add: T'_def vimage_def Collect_conj_eq continuous_on_diff continuous_on_setdist) |
|
735 have "S \<subseteq> S'" |
|
736 using S'_def \<open>S \<subseteq> U\<close> setdist_sing_in_set by fastforce |
|
737 have "T \<subseteq> T'" |
|
738 using T'_def \<open>T \<subseteq> U\<close> setdist_sing_in_set by fastforce |
|
739 have "S \<inter> T \<subseteq> W" "W \<subseteq> U" |
|
740 using \<open>S \<subseteq> U\<close> by (auto simp: W_def setdist_sing_in_set) |
|
741 have "(S \<inter> T) retract_of W" |
|
742 apply (rule AR_imp_absolute_retract [OF \<open>AR(S \<inter> T)\<close>]) |
|
743 apply (simp add: homeomorphic_refl) |
|
744 apply (rule closedin_subset_trans [of U]) |
|
745 apply (simp_all add: assms closedin_Int \<open>S \<inter> T \<subseteq> W\<close> \<open>W \<subseteq> U\<close>) |
|
746 done |
|
747 then obtain r0 |
|
748 where "S \<inter> T \<subseteq> W" and contr0: "continuous_on W r0" |
|
749 and "r0 ` W \<subseteq> S \<inter> T" |
|
750 and r0 [simp]: "\<And>x. x \<in> S \<inter> T \<Longrightarrow> r0 x = x" |
|
751 by (auto simp: retract_of_def retraction_def) |
|
752 have ST: "x \<in> W \<Longrightarrow> x \<in> S \<longleftrightarrow> x \<in> T" for x |
|
753 using setdist_eq_0_closedin \<open>S \<inter> T \<noteq> {}\<close> assms |
|
754 by (force simp: W_def setdist_sing_in_set) |
|
755 have "S' \<inter> T' = W" |
|
756 by (auto simp: S'_def T'_def W_def) |
|
757 then have cloUW: "closedin (top_of_set U) W" |
|
758 using closedin_Int US' UT' by blast |
|
759 define r where "r \<equiv> \<lambda>x. if x \<in> W then r0 x else x" |
|
760 have "r ` (W \<union> S) \<subseteq> S" "r ` (W \<union> T) \<subseteq> T" |
|
761 using \<open>r0 ` W \<subseteq> S \<inter> T\<close> r_def by auto |
|
762 have contr: "continuous_on (W \<union> (S \<union> T)) r" |
|
763 unfolding r_def |
|
764 proof (rule continuous_on_cases_local [OF _ _ contr0 continuous_on_id]) |
|
765 show "closedin (top_of_set (W \<union> (S \<union> T))) W" |
|
766 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 |
|
767 show "closedin (top_of_set (W \<union> (S \<union> T))) (S \<union> T)" |
|
768 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) |
|
769 show "\<And>x. x \<in> W \<and> x \<notin> W \<or> x \<in> S \<union> T \<and> x \<in> W \<Longrightarrow> r0 x = x" |
|
770 by (auto simp: ST) |
|
771 qed |
|
772 have cloUWS: "closedin (top_of_set U) (W \<union> S)" |
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773 by (simp add: cloUW assms closedin_Un) |
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774 obtain g where contg: "continuous_on U g" |
|
775 and "g ` U \<subseteq> S" and geqr: "\<And>x. x \<in> W \<union> S \<Longrightarrow> g x = r x" |
|
776 apply (rule AR_imp_absolute_extensor [OF \<open>AR S\<close> _ _ cloUWS]) |
|
777 apply (rule continuous_on_subset [OF contr]) |
|
778 using \<open>r ` (W \<union> S) \<subseteq> S\<close> apply auto |
|
779 done |
|
780 have cloUWT: "closedin (top_of_set U) (W \<union> T)" |
|
781 by (simp add: cloUW assms closedin_Un) |
|
782 obtain h where conth: "continuous_on U h" |
|
783 and "h ` U \<subseteq> T" and heqr: "\<And>x. x \<in> W \<union> T \<Longrightarrow> h x = r x" |
|
784 apply (rule AR_imp_absolute_extensor [OF \<open>AR T\<close> _ _ cloUWT]) |
|
785 apply (rule continuous_on_subset [OF contr]) |
|
786 using \<open>r ` (W \<union> T) \<subseteq> T\<close> apply auto |
|
787 done |
|
788 have "U = S' \<union> T'" |
|
789 by (force simp: S'_def T'_def) |
|
790 then have cont: "continuous_on U (\<lambda>x. if x \<in> S' then g x else h x)" |
|
791 apply (rule ssubst) |
|
792 apply (rule continuous_on_cases_local) |
|
793 using US' UT' \<open>S' \<inter> T' = W\<close> \<open>U = S' \<union> T'\<close> |
|
794 contg conth continuous_on_subset geqr heqr apply auto |
|
795 done |
|
796 have UST: "(\<lambda>x. if x \<in> S' then g x else h x) ` U \<subseteq> S \<union> T" |
|
797 using \<open>g ` U \<subseteq> S\<close> \<open>h ` U \<subseteq> T\<close> by auto |
|
798 show ?thesis |
|
799 apply (simp add: retract_of_def retraction_def \<open>S \<subseteq> U\<close> \<open>T \<subseteq> U\<close>) |
|
800 apply (rule_tac x="\<lambda>x. if x \<in> S' then g x else h x" in exI) |
|
801 apply (intro conjI cont UST) |
|
802 by (metis IntI ST Un_iff \<open>S \<subseteq> S'\<close> \<open>S' \<inter> T' = W\<close> \<open>T \<subseteq> T'\<close> subsetD geqr heqr r0 r_def) |
|
803 qed |
|
804 |
|
805 |
|
806 lemma AR_closed_Un_local: |
|
807 fixes S :: "'a::euclidean_space set" |
|
808 assumes STS: "closedin (top_of_set (S \<union> T)) S" |
|
809 and STT: "closedin (top_of_set (S \<union> T)) T" |
|
810 and "AR S" "AR T" "AR(S \<inter> T)" |
|
811 shows "AR(S \<union> T)" |
|
812 proof - |
|
813 have "C retract_of U" |
|
814 if hom: "S \<union> T homeomorphic C" and UC: "closedin (top_of_set U) C" |
|
815 for U and C :: "('a * real) set" |
|
816 proof - |
|
817 obtain f g where hom: "homeomorphism (S \<union> T) C f g" |
|
818 using hom by (force simp: homeomorphic_def) |
|
819 have US: "closedin (top_of_set U) (C \<inter> g -` S)" |
|
820 apply (rule closedin_trans [OF _ UC]) |
|
821 apply (rule continuous_closedin_preimage_gen [OF _ _ STS]) |
|
822 using hom homeomorphism_def apply blast |
|
823 apply (metis hom homeomorphism_def set_eq_subset) |
|
824 done |
|
825 have UT: "closedin (top_of_set U) (C \<inter> g -` T)" |
|
826 apply (rule closedin_trans [OF _ UC]) |
|
827 apply (rule continuous_closedin_preimage_gen [OF _ _ STT]) |
|
828 using hom homeomorphism_def apply blast |
|
829 apply (metis hom homeomorphism_def set_eq_subset) |
|
830 done |
|
831 have ARS: "AR (C \<inter> g -` S)" |
|
832 apply (rule AR_homeomorphic_AR [OF \<open>AR S\<close>]) |
|
833 apply (simp add: homeomorphic_def) |
|
834 apply (rule_tac x=g in exI) |
|
835 apply (rule_tac x=f in exI) |
|
836 using hom apply (auto simp: homeomorphism_def elim!: continuous_on_subset) |
|
837 apply (rule_tac x="f x" in image_eqI, auto) |
|
838 done |
|
839 have ART: "AR (C \<inter> g -` T)" |
|
840 apply (rule AR_homeomorphic_AR [OF \<open>AR T\<close>]) |
|
841 apply (simp add: homeomorphic_def) |
|
842 apply (rule_tac x=g in exI) |
|
843 apply (rule_tac x=f in exI) |
|
844 using hom apply (auto simp: homeomorphism_def elim!: continuous_on_subset) |
|
845 apply (rule_tac x="f x" in image_eqI, auto) |
|
846 done |
|
847 have ARI: "AR ((C \<inter> g -` S) \<inter> (C \<inter> g -` T))" |
|
848 apply (rule AR_homeomorphic_AR [OF \<open>AR (S \<inter> T)\<close>]) |
|
849 apply (simp add: homeomorphic_def) |
|
850 apply (rule_tac x=g in exI) |
|
851 apply (rule_tac x=f in exI) |
|
852 using hom |
|
853 apply (auto simp: homeomorphism_def elim!: continuous_on_subset) |
|
854 apply (rule_tac x="f x" in image_eqI, auto) |
|
855 done |
|
856 have "C = (C \<inter> g -` S) \<union> (C \<inter> g -` T)" |
|
857 using hom by (auto simp: homeomorphism_def) |
|
858 then show ?thesis |
|
859 by (metis AR_closed_Un_local_aux [OF US UT ARS ART ARI]) |
|
860 qed |
|
861 then show ?thesis |
|
862 by (force simp: AR_def) |
|
863 qed |
|
864 |
|
865 corollary AR_closed_Un: |
|
866 fixes S :: "'a::euclidean_space set" |
|
867 shows "\<lbrakk>closed S; closed T; AR S; AR T; AR (S \<inter> T)\<rbrakk> \<Longrightarrow> AR (S \<union> T)" |
|
868 by (metis AR_closed_Un_local_aux closed_closedin retract_of_UNIV subtopology_UNIV) |
|
869 |
|
870 text \<open>ANRs closed under union\<close> |
|
871 |
|
872 lemma ANR_closed_Un_local_aux: |
|
873 fixes U :: "'a::euclidean_space set" |
|
874 assumes US: "closedin (top_of_set U) S" |
|
875 and UT: "closedin (top_of_set U) T" |
|
876 and "ANR S" "ANR T" "ANR(S \<inter> T)" |
|
877 obtains V where "openin (top_of_set U) V" "(S \<union> T) retract_of V" |
|
878 proof (cases "S = {} \<or> T = {}") |
|
879 case True with assms that show ?thesis |
|
880 by (metis ANR_imp_neighbourhood_retract Un_commute inf_bot_right sup_inf_absorb) |
|
881 next |
|
882 case False |
|
883 then have [simp]: "S \<noteq> {}" "T \<noteq> {}" by auto |
|
884 have "S \<subseteq> U" "T \<subseteq> U" |
|
885 using assms by (auto simp: closedin_imp_subset) |
|
886 define S' where "S' \<equiv> {x \<in> U. setdist {x} S \<le> setdist {x} T}" |
|
887 define T' where "T' \<equiv> {x \<in> U. setdist {x} T \<le> setdist {x} S}" |
|
888 define W where "W \<equiv> {x \<in> U. setdist {x} S = setdist {x} T}" |
|
889 have cloUS': "closedin (top_of_set U) S'" |
|
890 using continuous_closedin_preimage [of U "\<lambda>x. setdist {x} S - setdist {x} T" "{..0}"] |
|
891 by (simp add: S'_def vimage_def Collect_conj_eq continuous_on_diff continuous_on_setdist) |
|
892 have cloUT': "closedin (top_of_set U) T'" |
|
893 using continuous_closedin_preimage [of U "\<lambda>x. setdist {x} T - setdist {x} S" "{..0}"] |
|
894 by (simp add: T'_def vimage_def Collect_conj_eq continuous_on_diff continuous_on_setdist) |
|
895 have "S \<subseteq> S'" |
|
896 using S'_def \<open>S \<subseteq> U\<close> setdist_sing_in_set by fastforce |
|
897 have "T \<subseteq> T'" |
|
898 using T'_def \<open>T \<subseteq> U\<close> setdist_sing_in_set by fastforce |
|
899 have "S' \<union> T' = U" |
|
900 by (auto simp: S'_def T'_def) |
|
901 have "W \<subseteq> S'" |
|
902 by (simp add: Collect_mono S'_def W_def) |
|
903 have "W \<subseteq> T'" |
|
904 by (simp add: Collect_mono T'_def W_def) |
|
905 have ST_W: "S \<inter> T \<subseteq> W" and "W \<subseteq> U" |
|
906 using \<open>S \<subseteq> U\<close> by (force simp: W_def setdist_sing_in_set)+ |
|
907 have "S' \<inter> T' = W" |
|
908 by (auto simp: S'_def T'_def W_def) |
|
909 then have cloUW: "closedin (top_of_set U) W" |
|
910 using closedin_Int cloUS' cloUT' by blast |
|
911 obtain W' W0 where "openin (top_of_set W) W'" |
|
912 and cloWW0: "closedin (top_of_set W) W0" |
|
913 and "S \<inter> T \<subseteq> W'" "W' \<subseteq> W0" |
|
914 and ret: "(S \<inter> T) retract_of W0" |
|
915 apply (rule ANR_imp_closed_neighbourhood_retract [OF \<open>ANR(S \<inter> T)\<close>]) |
|
916 apply (rule closedin_subset_trans [of U, OF _ ST_W \<open>W \<subseteq> U\<close>]) |
|
917 apply (blast intro: assms)+ |
|
918 done |
|
919 then obtain U0 where opeUU0: "openin (top_of_set U) U0" |
|
920 and U0: "S \<inter> T \<subseteq> U0" "U0 \<inter> W \<subseteq> W0" |
|
921 unfolding openin_open using \<open>W \<subseteq> U\<close> by blast |
|
922 have "W0 \<subseteq> U" |
|
923 using \<open>W \<subseteq> U\<close> cloWW0 closedin_subset by fastforce |
|
924 obtain r0 |
|
925 where "S \<inter> T \<subseteq> W0" and contr0: "continuous_on W0 r0" and "r0 ` W0 \<subseteq> S \<inter> T" |
|
926 and r0 [simp]: "\<And>x. x \<in> S \<inter> T \<Longrightarrow> r0 x = x" |
|
927 using ret by (force simp: retract_of_def retraction_def) |
|
928 have ST: "x \<in> W \<Longrightarrow> x \<in> S \<longleftrightarrow> x \<in> T" for x |
|
929 using assms by (auto simp: W_def setdist_sing_in_set dest!: setdist_eq_0_closedin) |
|
930 define r where "r \<equiv> \<lambda>x. if x \<in> W0 then r0 x else x" |
|
931 have "r ` (W0 \<union> S) \<subseteq> S" "r ` (W0 \<union> T) \<subseteq> T" |
|
932 using \<open>r0 ` W0 \<subseteq> S \<inter> T\<close> r_def by auto |
|
933 have contr: "continuous_on (W0 \<union> (S \<union> T)) r" |
|
934 unfolding r_def |
|
935 proof (rule continuous_on_cases_local [OF _ _ contr0 continuous_on_id]) |
|
936 show "closedin (top_of_set (W0 \<union> (S \<union> T))) W0" |
|
937 apply (rule closedin_subset_trans [of U]) |
|
938 using cloWW0 cloUW closedin_trans \<open>W0 \<subseteq> U\<close> \<open>S \<subseteq> U\<close> \<open>T \<subseteq> U\<close> apply blast+ |
|
939 done |
|
940 show "closedin (top_of_set (W0 \<union> (S \<union> T))) (S \<union> T)" |
|
941 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) |
|
942 show "\<And>x. x \<in> W0 \<and> x \<notin> W0 \<or> x \<in> S \<union> T \<and> x \<in> W0 \<Longrightarrow> r0 x = x" |
|
943 using ST cloWW0 closedin_subset by fastforce |
|
944 qed |
|
945 have cloS'WS: "closedin (top_of_set S') (W0 \<union> S)" |
|
946 by (meson closedin_subset_trans US cloUS' \<open>S \<subseteq> S'\<close> \<open>W \<subseteq> S'\<close> cloUW cloWW0 |
|
947 closedin_Un closedin_imp_subset closedin_trans) |
|
948 obtain W1 g where "W0 \<union> S \<subseteq> W1" and contg: "continuous_on W1 g" |
|
949 and opeSW1: "openin (top_of_set S') W1" |
|
950 and "g ` W1 \<subseteq> S" and geqr: "\<And>x. x \<in> W0 \<union> S \<Longrightarrow> g x = r x" |
|
951 apply (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> _ \<open>r ` (W0 \<union> S) \<subseteq> S\<close> cloS'WS]) |
|
952 apply (rule continuous_on_subset [OF contr], blast+) |
|
953 done |
|
954 have cloT'WT: "closedin (top_of_set T') (W0 \<union> T)" |
|
955 by (meson closedin_subset_trans UT cloUT' \<open>T \<subseteq> T'\<close> \<open>W \<subseteq> T'\<close> cloUW cloWW0 |
|
956 closedin_Un closedin_imp_subset closedin_trans) |
|
957 obtain W2 h where "W0 \<union> T \<subseteq> W2" and conth: "continuous_on W2 h" |
|
958 and opeSW2: "openin (top_of_set T') W2" |
|
959 and "h ` W2 \<subseteq> T" and heqr: "\<And>x. x \<in> W0 \<union> T \<Longrightarrow> h x = r x" |
|
960 apply (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> _ \<open>r ` (W0 \<union> T) \<subseteq> T\<close> cloT'WT]) |
|
961 apply (rule continuous_on_subset [OF contr], blast+) |
|
962 done |
|
963 have "S' \<inter> T' = W" |
|
964 by (force simp: S'_def T'_def W_def) |
|
965 obtain O1 O2 where "open O1" "W1 = S' \<inter> O1" "open O2" "W2 = T' \<inter> O2" |
|
966 using opeSW1 opeSW2 by (force simp: openin_open) |
|
967 show ?thesis |
|
968 proof |
|
969 have eq: "W1 - (W - U0) \<union> (W2 - (W - U0)) = |
|
970 ((U - T') \<inter> O1 \<union> (U - S') \<inter> O2 \<union> U \<inter> O1 \<inter> O2) - (W - U0)" |
|
971 using \<open>U0 \<inter> W \<subseteq> W0\<close> \<open>W0 \<union> S \<subseteq> W1\<close> \<open>W0 \<union> T \<subseteq> W2\<close> |
|
972 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>) |
|
973 show "openin (top_of_set U) (W1 - (W - U0) \<union> (W2 - (W - U0)))" |
|
974 apply (subst eq) |
|
975 apply (intro openin_Un openin_Int_open openin_diff closedin_diff cloUW opeUU0 cloUS' cloUT' \<open>open O1\<close> \<open>open O2\<close>, simp_all) |
|
976 done |
|
977 have cloW1: "closedin (top_of_set (W1 - (W - U0) \<union> (W2 - (W - U0)))) (W1 - (W - U0))" |
|
978 using cloUS' apply (simp add: closedin_closed) |
|
979 apply (erule ex_forward) |
|
980 using U0 \<open>W0 \<union> S \<subseteq> W1\<close> |
|
981 apply (auto simp: \<open>W1 = S' \<inter> O1\<close> \<open>W2 = T' \<inter> O2\<close> \<open>S' \<union> T' = U\<close> [symmetric]\<open>S' \<inter> T' = W\<close> [symmetric]) |
|
982 done |
|
983 have cloW2: "closedin (top_of_set (W1 - (W - U0) \<union> (W2 - (W - U0)))) (W2 - (W - U0))" |
|
984 using cloUT' apply (simp add: closedin_closed) |
|
985 apply (erule ex_forward) |
|
986 using U0 \<open>W0 \<union> T \<subseteq> W2\<close> |
|
987 apply (auto simp: \<open>W1 = S' \<inter> O1\<close> \<open>W2 = T' \<inter> O2\<close> \<open>S' \<union> T' = U\<close> [symmetric]\<open>S' \<inter> T' = W\<close> [symmetric]) |
|
988 done |
|
989 have *: "\<forall>x\<in>S \<union> T. (if x \<in> S' then g x else h x) = x" |
|
990 using ST \<open>S' \<inter> T' = W\<close> cloT'WT closedin_subset geqr heqr |
|
991 apply (auto simp: r_def, fastforce) |
|
992 using \<open>S \<subseteq> S'\<close> \<open>T \<subseteq> T'\<close> \<open>W0 \<union> S \<subseteq> W1\<close> \<open>W1 = S' \<inter> O1\<close> by auto |
|
993 have "\<exists>r. continuous_on (W1 - (W - U0) \<union> (W2 - (W - U0))) r \<and> |
|
994 r ` (W1 - (W - U0) \<union> (W2 - (W - U0))) \<subseteq> S \<union> T \<and> |
|
995 (\<forall>x\<in>S \<union> T. r x = x)" |
|
996 apply (rule_tac x = "\<lambda>x. if x \<in> S' then g x else h x" in exI) |
|
997 apply (intro conjI *) |
|
998 apply (rule continuous_on_cases_local |
|
999 [OF cloW1 cloW2 continuous_on_subset [OF contg] continuous_on_subset [OF conth]]) |
|
1000 using \<open>W1 = S' \<inter> O1\<close> \<open>W2 = T' \<inter> O2\<close> \<open>S' \<inter> T' = W\<close> |
|
1001 \<open>g ` W1 \<subseteq> S\<close> \<open>h ` W2 \<subseteq> T\<close> apply auto |
|
1002 using \<open>U0 \<inter> W \<subseteq> W0\<close> \<open>W0 \<union> S \<subseteq> W1\<close> apply (fastforce simp add: geqr heqr)+ |
|
1003 done |
|
1004 then show "S \<union> T retract_of W1 - (W - U0) \<union> (W2 - (W - U0))" |
|
1005 using \<open>W0 \<union> S \<subseteq> W1\<close> \<open>W0 \<union> T \<subseteq> W2\<close> ST opeUU0 U0 |
|
1006 by (auto simp: retract_of_def retraction_def) |
|
1007 qed |
|
1008 qed |
|
1009 |
|
1010 |
|
1011 lemma ANR_closed_Un_local: |
|
1012 fixes S :: "'a::euclidean_space set" |
|
1013 assumes STS: "closedin (top_of_set (S \<union> T)) S" |
|
1014 and STT: "closedin (top_of_set (S \<union> T)) T" |
|
1015 and "ANR S" "ANR T" "ANR(S \<inter> T)" |
|
1016 shows "ANR(S \<union> T)" |
|
1017 proof - |
|
1018 have "\<exists>T. openin (top_of_set U) T \<and> C retract_of T" |
|
1019 if hom: "S \<union> T homeomorphic C" and UC: "closedin (top_of_set U) C" |
|
1020 for U and C :: "('a * real) set" |
|
1021 proof - |
|
1022 obtain f g where hom: "homeomorphism (S \<union> T) C f g" |
|
1023 using hom by (force simp: homeomorphic_def) |
|
1024 have US: "closedin (top_of_set U) (C \<inter> g -` S)" |
|
1025 apply (rule closedin_trans [OF _ UC]) |
|
1026 apply (rule continuous_closedin_preimage_gen [OF _ _ STS]) |
|
1027 using hom [unfolded homeomorphism_def] apply blast |
|
1028 apply (metis hom homeomorphism_def set_eq_subset) |
|
1029 done |
|
1030 have UT: "closedin (top_of_set U) (C \<inter> g -` T)" |
|
1031 apply (rule closedin_trans [OF _ UC]) |
|
1032 apply (rule continuous_closedin_preimage_gen [OF _ _ STT]) |
|
1033 using hom [unfolded homeomorphism_def] apply blast |
|
1034 apply (metis hom homeomorphism_def set_eq_subset) |
|
1035 done |
|
1036 have ANRS: "ANR (C \<inter> g -` S)" |
|
1037 apply (rule ANR_homeomorphic_ANR [OF \<open>ANR S\<close>]) |
|
1038 apply (simp add: homeomorphic_def) |
|
1039 apply (rule_tac x=g in exI) |
|
1040 apply (rule_tac x=f in exI) |
|
1041 using hom apply (auto simp: homeomorphism_def elim!: continuous_on_subset) |
|
1042 apply (rule_tac x="f x" in image_eqI, auto) |
|
1043 done |
|
1044 have ANRT: "ANR (C \<inter> g -` T)" |
|
1045 apply (rule ANR_homeomorphic_ANR [OF \<open>ANR T\<close>]) |
|
1046 apply (simp add: homeomorphic_def) |
|
1047 apply (rule_tac x=g in exI) |
|
1048 apply (rule_tac x=f in exI) |
|
1049 using hom apply (auto simp: homeomorphism_def elim!: continuous_on_subset) |
|
1050 apply (rule_tac x="f x" in image_eqI, auto) |
|
1051 done |
|
1052 have ANRI: "ANR ((C \<inter> g -` S) \<inter> (C \<inter> g -` T))" |
|
1053 apply (rule ANR_homeomorphic_ANR [OF \<open>ANR (S \<inter> T)\<close>]) |
|
1054 apply (simp add: homeomorphic_def) |
|
1055 apply (rule_tac x=g in exI) |
|
1056 apply (rule_tac x=f in exI) |
|
1057 using hom |
|
1058 apply (auto simp: homeomorphism_def elim!: continuous_on_subset) |
|
1059 apply (rule_tac x="f x" in image_eqI, auto) |
|
1060 done |
|
1061 have "C = (C \<inter> g -` S) \<union> (C \<inter> g -` T)" |
|
1062 using hom by (auto simp: homeomorphism_def) |
|
1063 then show ?thesis |
|
1064 by (metis ANR_closed_Un_local_aux [OF US UT ANRS ANRT ANRI]) |
|
1065 qed |
|
1066 then show ?thesis |
|
1067 by (auto simp: ANR_def) |
|
1068 qed |
|
1069 |
|
1070 corollary ANR_closed_Un: |
|
1071 fixes S :: "'a::euclidean_space set" |
|
1072 shows "\<lbrakk>closed S; closed T; ANR S; ANR T; ANR (S \<inter> T)\<rbrakk> \<Longrightarrow> ANR (S \<union> T)" |
|
1073 by (simp add: ANR_closed_Un_local closedin_def diff_eq open_Compl openin_open_Int) |
|
1074 |
|
1075 lemma ANR_openin: |
|
1076 fixes S :: "'a::euclidean_space set" |
|
1077 assumes "ANR T" and opeTS: "openin (top_of_set T) S" |
|
1078 shows "ANR S" |
|
1079 proof (clarsimp simp only: ANR_eq_absolute_neighbourhood_extensor) |
|
1080 fix f :: "'a \<times> real \<Rightarrow> 'a" and U C |
|
1081 assume contf: "continuous_on C f" and fim: "f ` C \<subseteq> S" |
|
1082 and cloUC: "closedin (top_of_set U) C" |
|
1083 have "f ` C \<subseteq> T" |
|
1084 using fim opeTS openin_imp_subset by blast |
|
1085 obtain W g where "C \<subseteq> W" |
|
1086 and UW: "openin (top_of_set U) W" |
|
1087 and contg: "continuous_on W g" |
|
1088 and gim: "g ` W \<subseteq> T" |
|
1089 and geq: "\<And>x. x \<in> C \<Longrightarrow> g x = f x" |
|
1090 apply (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> contf \<open>f ` C \<subseteq> T\<close> cloUC]) |
|
1091 using fim by auto |
|
1092 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)" |
|
1093 proof (intro exI conjI) |
|
1094 show "C \<subseteq> W \<inter> g -` S" |
|
1095 using \<open>C \<subseteq> W\<close> fim geq by blast |
|
1096 show "openin (top_of_set U) (W \<inter> g -` S)" |
|
1097 by (metis (mono_tags, lifting) UW contg continuous_openin_preimage gim opeTS openin_trans) |
|
1098 show "continuous_on (W \<inter> g -` S) g" |
|
1099 by (blast intro: continuous_on_subset [OF contg]) |
|
1100 show "g ` (W \<inter> g -` S) \<subseteq> S" |
|
1101 using gim by blast |
|
1102 show "\<forall>x\<in>C. g x = f x" |
|
1103 using geq by blast |
|
1104 qed |
|
1105 qed |
|
1106 |
|
1107 lemma ENR_openin: |
|
1108 fixes S :: "'a::euclidean_space set" |
|
1109 assumes "ENR T" and opeTS: "openin (top_of_set T) S" |
|
1110 shows "ENR S" |
|
1111 using assms apply (simp add: ENR_ANR) |
|
1112 using ANR_openin locally_open_subset by blast |
|
1113 |
|
1114 lemma ANR_neighborhood_retract: |
|
1115 fixes S :: "'a::euclidean_space set" |
|
1116 assumes "ANR U" "S retract_of T" "openin (top_of_set U) T" |
|
1117 shows "ANR S" |
|
1118 using ANR_openin ANR_retract_of_ANR assms by blast |
|
1119 |
|
1120 lemma ENR_neighborhood_retract: |
|
1121 fixes S :: "'a::euclidean_space set" |
|
1122 assumes "ENR U" "S retract_of T" "openin (top_of_set U) T" |
|
1123 shows "ENR S" |
|
1124 using ENR_openin ENR_retract_of_ENR assms by blast |
|
1125 |
|
1126 lemma ANR_rel_interior: |
|
1127 fixes S :: "'a::euclidean_space set" |
|
1128 shows "ANR S \<Longrightarrow> ANR(rel_interior S)" |
|
1129 by (blast intro: ANR_openin openin_set_rel_interior) |
|
1130 |
|
1131 lemma ANR_delete: |
|
1132 fixes S :: "'a::euclidean_space set" |
|
1133 shows "ANR S \<Longrightarrow> ANR(S - {a})" |
|
1134 by (blast intro: ANR_openin openin_delete openin_subtopology_self) |
|
1135 |
|
1136 lemma ENR_rel_interior: |
|
1137 fixes S :: "'a::euclidean_space set" |
|
1138 shows "ENR S \<Longrightarrow> ENR(rel_interior S)" |
|
1139 by (blast intro: ENR_openin openin_set_rel_interior) |
|
1140 |
|
1141 lemma ENR_delete: |
|
1142 fixes S :: "'a::euclidean_space set" |
|
1143 shows "ENR S \<Longrightarrow> ENR(S - {a})" |
|
1144 by (blast intro: ENR_openin openin_delete openin_subtopology_self) |
|
1145 |
|
1146 lemma open_imp_ENR: "open S \<Longrightarrow> ENR S" |
|
1147 using ENR_def by blast |
|
1148 |
|
1149 lemma open_imp_ANR: |
|
1150 fixes S :: "'a::euclidean_space set" |
|
1151 shows "open S \<Longrightarrow> ANR S" |
|
1152 by (simp add: ENR_imp_ANR open_imp_ENR) |
|
1153 |
|
1154 lemma ANR_ball [iff]: |
|
1155 fixes a :: "'a::euclidean_space" |
|
1156 shows "ANR(ball a r)" |
|
1157 by (simp add: convex_imp_ANR) |
|
1158 |
|
1159 lemma ENR_ball [iff]: "ENR(ball a r)" |
|
1160 by (simp add: open_imp_ENR) |
|
1161 |
|
1162 lemma AR_ball [simp]: |
|
1163 fixes a :: "'a::euclidean_space" |
|
1164 shows "AR(ball a r) \<longleftrightarrow> 0 < r" |
|
1165 by (auto simp: AR_ANR convex_imp_contractible) |
|
1166 |
|
1167 lemma ANR_cball [iff]: |
|
1168 fixes a :: "'a::euclidean_space" |
|
1169 shows "ANR(cball a r)" |
|
1170 by (simp add: convex_imp_ANR) |
|
1171 |
|
1172 lemma ENR_cball: |
|
1173 fixes a :: "'a::euclidean_space" |
|
1174 shows "ENR(cball a r)" |
|
1175 using ENR_convex_closed by blast |
|
1176 |
|
1177 lemma AR_cball [simp]: |
|
1178 fixes a :: "'a::euclidean_space" |
|
1179 shows "AR(cball a r) \<longleftrightarrow> 0 \<le> r" |
|
1180 by (auto simp: AR_ANR convex_imp_contractible) |
|
1181 |
|
1182 lemma ANR_box [iff]: |
|
1183 fixes a :: "'a::euclidean_space" |
|
1184 shows "ANR(cbox a b)" "ANR(box a b)" |
|
1185 by (auto simp: convex_imp_ANR open_imp_ANR) |
|
1186 |
|
1187 lemma ENR_box [iff]: |
|
1188 fixes a :: "'a::euclidean_space" |
|
1189 shows "ENR(cbox a b)" "ENR(box a b)" |
|
1190 apply (simp add: ENR_convex_closed closed_cbox) |
|
1191 by (simp add: open_box open_imp_ENR) |
|
1192 |
|
1193 lemma AR_box [simp]: |
|
1194 "AR(cbox a b) \<longleftrightarrow> cbox a b \<noteq> {}" "AR(box a b) \<longleftrightarrow> box a b \<noteq> {}" |
|
1195 by (auto simp: AR_ANR convex_imp_contractible) |
|
1196 |
|
1197 lemma ANR_interior: |
|
1198 fixes S :: "'a::euclidean_space set" |
|
1199 shows "ANR(interior S)" |
|
1200 by (simp add: open_imp_ANR) |
|
1201 |
|
1202 lemma ENR_interior: |
|
1203 fixes S :: "'a::euclidean_space set" |
|
1204 shows "ENR(interior S)" |
|
1205 by (simp add: open_imp_ENR) |
|
1206 |
|
1207 lemma AR_imp_contractible: |
|
1208 fixes S :: "'a::euclidean_space set" |
|
1209 shows "AR S \<Longrightarrow> contractible S" |
|
1210 by (simp add: AR_ANR) |
|
1211 |
|
1212 lemma ENR_imp_locally_compact: |
|
1213 fixes S :: "'a::euclidean_space set" |
|
1214 shows "ENR S \<Longrightarrow> locally compact S" |
|
1215 by (simp add: ENR_ANR) |
|
1216 |
|
1217 lemma ANR_imp_locally_path_connected: |
|
1218 fixes S :: "'a::euclidean_space set" |
|
1219 assumes "ANR S" |
|
1220 shows "locally path_connected S" |
|
1221 proof - |
|
1222 obtain U and T :: "('a \<times> real) set" |
|
1223 where "convex U" "U \<noteq> {}" |
|
1224 and UT: "closedin (top_of_set U) T" |
|
1225 and "S homeomorphic T" |
|
1226 apply (rule homeomorphic_closedin_convex [of S]) |
|
1227 using aff_dim_le_DIM [of S] apply auto |
|
1228 done |
|
1229 then have "locally path_connected T" |
|
1230 by (meson ANR_imp_absolute_neighbourhood_retract |
|
1231 assms convex_imp_locally_path_connected locally_open_subset retract_of_locally_path_connected) |
|
1232 then have S: "locally path_connected S" |
|
1233 if "openin (top_of_set U) V" "T retract_of V" "U \<noteq> {}" for V |
|
1234 using \<open>S homeomorphic T\<close> homeomorphic_locally homeomorphic_path_connectedness by blast |
|
1235 show ?thesis |
|
1236 using assms |
|
1237 apply (clarsimp simp: ANR_def) |
|
1238 apply (drule_tac x=U in spec) |
|
1239 apply (drule_tac x=T in spec) |
|
1240 using \<open>S homeomorphic T\<close> \<open>U \<noteq> {}\<close> UT apply (blast intro: S) |
|
1241 done |
|
1242 qed |
|
1243 |
|
1244 lemma ANR_imp_locally_connected: |
|
1245 fixes S :: "'a::euclidean_space set" |
|
1246 assumes "ANR S" |
|
1247 shows "locally connected S" |
|
1248 using locally_path_connected_imp_locally_connected ANR_imp_locally_path_connected assms by auto |
|
1249 |
|
1250 lemma AR_imp_locally_path_connected: |
|
1251 fixes S :: "'a::euclidean_space set" |
|
1252 assumes "AR S" |
|
1253 shows "locally path_connected S" |
|
1254 by (simp add: ANR_imp_locally_path_connected AR_imp_ANR assms) |
|
1255 |
|
1256 lemma AR_imp_locally_connected: |
|
1257 fixes S :: "'a::euclidean_space set" |
|
1258 assumes "AR S" |
|
1259 shows "locally connected S" |
|
1260 using ANR_imp_locally_connected AR_ANR assms by blast |
|
1261 |
|
1262 lemma ENR_imp_locally_path_connected: |
|
1263 fixes S :: "'a::euclidean_space set" |
|
1264 assumes "ENR S" |
|
1265 shows "locally path_connected S" |
|
1266 by (simp add: ANR_imp_locally_path_connected ENR_imp_ANR assms) |
|
1267 |
|
1268 lemma ENR_imp_locally_connected: |
|
1269 fixes S :: "'a::euclidean_space set" |
|
1270 assumes "ENR S" |
|
1271 shows "locally connected S" |
|
1272 using ANR_imp_locally_connected ENR_ANR assms by blast |
|
1273 |
|
1274 lemma ANR_Times: |
|
1275 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
|
1276 assumes "ANR S" "ANR T" shows "ANR(S \<times> T)" |
|
1277 proof (clarsimp simp only: ANR_eq_absolute_neighbourhood_extensor) |
|
1278 fix f :: " ('a \<times> 'b) \<times> real \<Rightarrow> 'a \<times> 'b" and U C |
|
1279 assume "continuous_on C f" and fim: "f ` C \<subseteq> S \<times> T" |
|
1280 and cloUC: "closedin (top_of_set U) C" |
|
1281 have contf1: "continuous_on C (fst \<circ> f)" |
|
1282 by (simp add: \<open>continuous_on C f\<close> continuous_on_fst) |
|
1283 obtain W1 g where "C \<subseteq> W1" |
|
1284 and UW1: "openin (top_of_set U) W1" |
|
1285 and contg: "continuous_on W1 g" |
|
1286 and gim: "g ` W1 \<subseteq> S" |
|
1287 and geq: "\<And>x. x \<in> C \<Longrightarrow> g x = (fst \<circ> f) x" |
|
1288 apply (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR S\<close> contf1 _ cloUC]) |
|
1289 using fim apply auto |
|
1290 done |
|
1291 have contf2: "continuous_on C (snd \<circ> f)" |
|
1292 by (simp add: \<open>continuous_on C f\<close> continuous_on_snd) |
|
1293 obtain W2 h where "C \<subseteq> W2" |
|
1294 and UW2: "openin (top_of_set U) W2" |
|
1295 and conth: "continuous_on W2 h" |
|
1296 and him: "h ` W2 \<subseteq> T" |
|
1297 and heq: "\<And>x. x \<in> C \<Longrightarrow> h x = (snd \<circ> f) x" |
|
1298 apply (rule ANR_imp_absolute_neighbourhood_extensor [OF \<open>ANR T\<close> contf2 _ cloUC]) |
|
1299 using fim apply auto |
|
1300 done |
|
1301 show "\<exists>V g. C \<subseteq> V \<and> |
|
1302 openin (top_of_set U) V \<and> |
|
1303 continuous_on V g \<and> g ` V \<subseteq> S \<times> T \<and> (\<forall>x\<in>C. g x = f x)" |
|
1304 proof (intro exI conjI) |
|
1305 show "C \<subseteq> W1 \<inter> W2" |
|
1306 by (simp add: \<open>C \<subseteq> W1\<close> \<open>C \<subseteq> W2\<close>) |
|
1307 show "openin (top_of_set U) (W1 \<inter> W2)" |
|
1308 by (simp add: UW1 UW2 openin_Int) |
|
1309 show "continuous_on (W1 \<inter> W2) (\<lambda>x. (g x, h x))" |
|
1310 by (metis (no_types) contg conth continuous_on_Pair continuous_on_subset inf_commute inf_le1) |
|
1311 show "(\<lambda>x. (g x, h x)) ` (W1 \<inter> W2) \<subseteq> S \<times> T" |
|
1312 using gim him by blast |
|
1313 show "(\<forall>x\<in>C. (g x, h x) = f x)" |
|
1314 using geq heq by auto |
|
1315 qed |
|
1316 qed |
|
1317 |
|
1318 lemma AR_Times: |
|
1319 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
|
1320 assumes "AR S" "AR T" shows "AR(S \<times> T)" |
|
1321 using assms by (simp add: AR_ANR ANR_Times contractible_Times) |
|
1322 |
|
1323 subsection\<^marker>\<open>tag unimportant\<close>\<open>Retracts and intervals in ordered euclidean space\<close> |
|
1324 |
|
1325 lemma ANR_interval [iff]: |
|
1326 fixes a :: "'a::ordered_euclidean_space" |
|
1327 shows "ANR{a..b}" |
|
1328 by (simp add: interval_cbox) |
|
1329 |
|
1330 lemma ENR_interval [iff]: |
|
1331 fixes a :: "'a::ordered_euclidean_space" |
|
1332 shows "ENR{a..b}" |
|
1333 by (auto simp: interval_cbox) |
|
1334 |
|
1335 subsection \<open>More advanced properties of ANRs and ENRs\<close> |
|
1336 |
|
1337 lemma ENR_rel_frontier_convex: |
|
1338 fixes S :: "'a::euclidean_space set" |
|
1339 assumes "bounded S" "convex S" |
|
1340 shows "ENR(rel_frontier S)" |
|
1341 proof (cases "S = {}") |
|
1342 case True then show ?thesis |
|
1343 by simp |
|
1344 next |
|
1345 case False |
|
1346 with assms have "rel_interior S \<noteq> {}" |
|
1347 by (simp add: rel_interior_eq_empty) |
|
1348 then obtain a where a: "a \<in> rel_interior S" |
|
1349 by auto |
|
1350 have ahS: "affine hull S - {a} \<subseteq> {x. closest_point (affine hull S) x \<noteq> a}" |
|
1351 by (auto simp: closest_point_self) |
|
1352 have "rel_frontier S retract_of affine hull S - {a}" |
|
1353 by (simp add: assms a rel_frontier_retract_of_punctured_affine_hull) |
|
1354 also have "\<dots> retract_of {x. closest_point (affine hull S) x \<noteq> a}" |
|
1355 apply (simp add: retract_of_def retraction_def ahS) |
|
1356 apply (rule_tac x="closest_point (affine hull S)" in exI) |
|
1357 apply (auto simp: False closest_point_self affine_imp_convex closest_point_in_set continuous_on_closest_point) |
|
1358 done |
|
1359 finally have "rel_frontier S retract_of {x. closest_point (affine hull S) x \<noteq> a}" . |
|
1360 moreover have "openin (top_of_set UNIV) (UNIV \<inter> closest_point (affine hull S) -` (- {a}))" |
|
1361 apply (rule continuous_openin_preimage_gen) |
|
1362 apply (auto simp: False affine_imp_convex continuous_on_closest_point) |
|
1363 done |
|
1364 ultimately show ?thesis |
|
1365 unfolding ENR_def |
|
1366 apply (rule_tac x = "closest_point (affine hull S) -` (- {a})" in exI) |
|
1367 apply (simp add: vimage_def) |
|
1368 done |
|
1369 qed |
|
1370 |
|
1371 lemma ANR_rel_frontier_convex: |
|
1372 fixes S :: "'a::euclidean_space set" |
|
1373 assumes "bounded S" "convex S" |
|
1374 shows "ANR(rel_frontier S)" |
|
1375 by (simp add: ENR_imp_ANR ENR_rel_frontier_convex assms) |
|
1376 |
|
1377 lemma ENR_closedin_Un_local: |
|
1378 fixes S :: "'a::euclidean_space set" |
|
1379 shows "\<lbrakk>ENR S; ENR T; ENR(S \<inter> T); |
|
1380 closedin (top_of_set (S \<union> T)) S; closedin (top_of_set (S \<union> T)) T\<rbrakk> |
|
1381 \<Longrightarrow> ENR(S \<union> T)" |
|
1382 by (simp add: ENR_ANR ANR_closed_Un_local locally_compact_closedin_Un) |
|
1383 |
|
1384 lemma ENR_closed_Un: |
|
1385 fixes S :: "'a::euclidean_space set" |
|
1386 shows "\<lbrakk>closed S; closed T; ENR S; ENR T; ENR(S \<inter> T)\<rbrakk> \<Longrightarrow> ENR(S \<union> T)" |
|
1387 by (auto simp: closed_subset ENR_closedin_Un_local) |
|
1388 |
|
1389 lemma absolute_retract_Un: |
|
1390 fixes S :: "'a::euclidean_space set" |
|
1391 shows "\<lbrakk>S retract_of UNIV; T retract_of UNIV; (S \<inter> T) retract_of UNIV\<rbrakk> |
|
1392 \<Longrightarrow> (S \<union> T) retract_of UNIV" |
|
1393 by (meson AR_closed_Un_local_aux closed_subset retract_of_UNIV retract_of_imp_subset) |
|
1394 |
|
1395 lemma retract_from_Un_Int: |
|
1396 fixes S :: "'a::euclidean_space set" |
|
1397 assumes clS: "closedin (top_of_set (S \<union> T)) S" |
|
1398 and clT: "closedin (top_of_set (S \<union> T)) T" |
|
1399 and Un: "(S \<union> T) retract_of U" and Int: "(S \<inter> T) retract_of T" |
|
1400 shows "S retract_of U" |
|
1401 proof - |
|
1402 obtain r where r: "continuous_on T r" "r ` T \<subseteq> S \<inter> T" "\<forall>x\<in>S \<inter> T. r x = x" |
|
1403 using Int by (auto simp: retraction_def retract_of_def) |
|
1404 have "S retract_of S \<union> T" |
|
1405 unfolding retraction_def retract_of_def |
|
1406 proof (intro exI conjI) |
|
1407 show "continuous_on (S \<union> T) (\<lambda>x. if x \<in> S then x else r x)" |
|
1408 apply (rule continuous_on_cases_local [OF clS clT]) |
|
1409 using r by (auto simp: continuous_on_id) |
|
1410 qed (use r in auto) |
|
1411 also have "\<dots> retract_of U" |
|
1412 by (rule Un) |
|
1413 finally show ?thesis . |
|
1414 qed |
|
1415 |
|
1416 lemma AR_from_Un_Int_local: |
|
1417 fixes S :: "'a::euclidean_space set" |
|
1418 assumes clS: "closedin (top_of_set (S \<union> T)) S" |
|
1419 and clT: "closedin (top_of_set (S \<union> T)) T" |
|
1420 and Un: "AR(S \<union> T)" and Int: "AR(S \<inter> T)" |
|
1421 shows "AR S" |
|
1422 apply (rule AR_retract_of_AR [OF Un]) |
|
1423 by (meson AR_imp_retract clS clT closedin_closed_subset local.Int retract_from_Un_Int retract_of_refl sup_ge2) |
|
1424 |
|
1425 lemma AR_from_Un_Int_local': |
|
1426 fixes S :: "'a::euclidean_space set" |
|
1427 assumes "closedin (top_of_set (S \<union> T)) S" |
|
1428 and "closedin (top_of_set (S \<union> T)) T" |
|
1429 and "AR(S \<union> T)" "AR(S \<inter> T)" |
|
1430 shows "AR T" |
|
1431 using AR_from_Un_Int_local [of T S] assms by (simp add: Un_commute Int_commute) |
|
1432 |
|
1433 lemma AR_from_Un_Int: |
|
1434 fixes S :: "'a::euclidean_space set" |
|
1435 assumes clo: "closed S" "closed T" and Un: "AR(S \<union> T)" and Int: "AR(S \<inter> T)" |
|
1436 shows "AR S" |
|
1437 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) |
|
1438 |
|
1439 lemma ANR_from_Un_Int_local: |
|
1440 fixes S :: "'a::euclidean_space set" |
|
1441 assumes clS: "closedin (top_of_set (S \<union> T)) S" |
|
1442 and clT: "closedin (top_of_set (S \<union> T)) T" |
|
1443 and Un: "ANR(S \<union> T)" and Int: "ANR(S \<inter> T)" |
|
1444 shows "ANR S" |
|
1445 proof - |
|
1446 obtain V where clo: "closedin (top_of_set (S \<union> T)) (S \<inter> T)" |
|
1447 and ope: "openin (top_of_set (S \<union> T)) V" |
|
1448 and ret: "S \<inter> T retract_of V" |
|
1449 using ANR_imp_neighbourhood_retract [OF Int] by (metis clS clT closedin_Int) |
|
1450 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" |
|
1451 by (auto simp: retraction_def retract_of_def) |
|
1452 have Vsub: "V \<subseteq> S \<union> T" |
|
1453 by (meson ope openin_contains_cball) |
|
1454 have Vsup: "S \<inter> T \<subseteq> V" |
|
1455 by (simp add: retract_of_imp_subset ret) |
|
1456 then have eq: "S \<union> V = ((S \<union> T) - T) \<union> V" |
|
1457 by auto |
|
1458 have eq': "S \<union> V = S \<union> (V \<inter> T)" |
|
1459 using Vsub by blast |
|
1460 have "continuous_on (S \<union> V \<inter> T) (\<lambda>x. if x \<in> S then x else r x)" |
|
1461 proof (rule continuous_on_cases_local) |
|
1462 show "closedin (top_of_set (S \<union> V \<inter> T)) S" |
|
1463 using clS closedin_subset_trans inf.boundedE by blast |
|
1464 show "closedin (top_of_set (S \<union> V \<inter> T)) (V \<inter> T)" |
|
1465 using clT Vsup by (auto simp: closedin_closed) |
|
1466 show "continuous_on (V \<inter> T) r" |
|
1467 by (meson Int_lower1 continuous_on_subset r) |
|
1468 qed (use req continuous_on_id in auto) |
|
1469 with rim have "S retract_of S \<union> V" |
|
1470 unfolding retraction_def retract_of_def |
|
1471 apply (rule_tac x="\<lambda>x. if x \<in> S then x else r x" in exI) |
|
1472 apply (auto simp: eq') |
|
1473 done |
|
1474 then show ?thesis |
|
1475 using ANR_neighborhood_retract [OF Un] |
|
1476 using \<open>S \<union> V = S \<union> T - T \<union> V\<close> clT ope by fastforce |
|
1477 qed |
|
1478 |
|
1479 lemma ANR_from_Un_Int: |
|
1480 fixes S :: "'a::euclidean_space set" |
|
1481 assumes clo: "closed S" "closed T" and Un: "ANR(S \<union> T)" and Int: "ANR(S \<inter> T)" |
|
1482 shows "ANR S" |
|
1483 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) |
|
1484 |
|
1485 lemma ANR_finite_Union_convex_closed: |
|
1486 fixes \<T> :: "'a::euclidean_space set set" |
|
1487 assumes \<T>: "finite \<T>" and clo: "\<And>C. C \<in> \<T> \<Longrightarrow> closed C" and con: "\<And>C. C \<in> \<T> \<Longrightarrow> convex C" |
|
1488 shows "ANR(\<Union>\<T>)" |
|
1489 proof - |
|
1490 have "ANR(\<Union>\<T>)" if "card \<T> < n" for n |
|
1491 using assms that |
|
1492 proof (induction n arbitrary: \<T>) |
|
1493 case 0 then show ?case by simp |
|
1494 next |
|
1495 case (Suc n) |
|
1496 have "ANR(\<Union>\<U>)" if "finite \<U>" "\<U> \<subseteq> \<T>" for \<U> |
|
1497 using that |
|
1498 proof (induction \<U>) |
|
1499 case empty |
|
1500 then show ?case by simp |
|
1501 next |
|
1502 case (insert C \<U>) |
|
1503 have "ANR (C \<union> \<Union>\<U>)" |
|
1504 proof (rule ANR_closed_Un) |
|
1505 show "ANR (C \<inter> \<Union>\<U>)" |
|
1506 unfolding Int_Union |
|
1507 proof (rule Suc) |
|
1508 show "finite ((\<inter>) C ` \<U>)" |
|
1509 by (simp add: insert.hyps(1)) |
|
1510 show "\<And>Ca. Ca \<in> (\<inter>) C ` \<U> \<Longrightarrow> closed Ca" |
|
1511 by (metis (no_types, hide_lams) Suc.prems(2) closed_Int subsetD imageE insert.prems insertI1 insertI2) |
|
1512 show "\<And>Ca. Ca \<in> (\<inter>) C ` \<U> \<Longrightarrow> convex Ca" |
|
1513 by (metis (mono_tags, lifting) Suc.prems(3) convex_Int imageE insert.prems insert_subset subsetCE) |
|
1514 show "card ((\<inter>) C ` \<U>) < n" |
|
1515 proof - |
|
1516 have "card \<T> \<le> n" |
|
1517 by (meson Suc.prems(4) not_less not_less_eq) |
|
1518 then show ?thesis |
|
1519 by (metis Suc.prems(1) card_image_le card_seteq insert.hyps insert.prems insert_subset le_trans not_less) |
|
1520 qed |
|
1521 qed |
|
1522 show "closed (\<Union>\<U>)" |
|
1523 using Suc.prems(2) insert.hyps(1) insert.prems by blast |
|
1524 qed (use Suc.prems convex_imp_ANR insert.prems insert.IH in auto) |
|
1525 then show ?case |
|
1526 by simp |
|
1527 qed |
|
1528 then show ?case |
|
1529 using Suc.prems(1) by blast |
|
1530 qed |
|
1531 then show ?thesis |
|
1532 by blast |
|
1533 qed |
|
1534 |
|
1535 |
|
1536 lemma finite_imp_ANR: |
|
1537 fixes S :: "'a::euclidean_space set" |
|
1538 assumes "finite S" |
|
1539 shows "ANR S" |
|
1540 proof - |
|
1541 have "ANR(\<Union>x \<in> S. {x})" |
|
1542 by (blast intro: ANR_finite_Union_convex_closed assms) |
|
1543 then show ?thesis |
|
1544 by simp |
|
1545 qed |
|
1546 |
|
1547 lemma ANR_insert: |
|
1548 fixes S :: "'a::euclidean_space set" |
|
1549 assumes "ANR S" "closed S" |
|
1550 shows "ANR(insert a S)" |
|
1551 by (metis ANR_closed_Un ANR_empty ANR_singleton Diff_disjoint Diff_insert_absorb assms closed_singleton insert_absorb insert_is_Un) |
|
1552 |
|
1553 lemma ANR_path_component_ANR: |
|
1554 fixes S :: "'a::euclidean_space set" |
|
1555 shows "ANR S \<Longrightarrow> ANR(path_component_set S x)" |
|
1556 using ANR_imp_locally_path_connected ANR_openin openin_path_component_locally_path_connected by blast |
|
1557 |
|
1558 lemma ANR_connected_component_ANR: |
|
1559 fixes S :: "'a::euclidean_space set" |
|
1560 shows "ANR S \<Longrightarrow> ANR(connected_component_set S x)" |
|
1561 by (metis ANR_openin openin_connected_component_locally_connected ANR_imp_locally_connected) |
|
1562 |
|
1563 lemma ANR_component_ANR: |
|
1564 fixes S :: "'a::euclidean_space set" |
|
1565 assumes "ANR S" "c \<in> components S" |
|
1566 shows "ANR c" |
|
1567 by (metis ANR_connected_component_ANR assms componentsE) |
|
1568 |
|
1569 subsection\<open>Original ANR material, now for ENRs\<close> |
|
1570 |
|
1571 lemma ENR_bounded: |
|
1572 fixes S :: "'a::euclidean_space set" |
|
1573 assumes "bounded S" |
|
1574 shows "ENR S \<longleftrightarrow> (\<exists>U. open U \<and> bounded U \<and> S retract_of U)" |
|
1575 (is "?lhs = ?rhs") |
|
1576 proof |
|
1577 obtain r where "0 < r" and r: "S \<subseteq> ball 0 r" |
|
1578 using bounded_subset_ballD assms by blast |
|
1579 assume ?lhs |
|
1580 then show ?rhs |
|
1581 apply (clarsimp simp: ENR_def) |
|
1582 apply (rule_tac x="ball 0 r \<inter> U" in exI, auto) |
|
1583 using r retract_of_imp_subset retract_of_subset by fastforce |
|
1584 next |
|
1585 assume ?rhs |
|
1586 then show ?lhs |
|
1587 using ENR_def by blast |
|
1588 qed |
|
1589 |
|
1590 lemma absolute_retract_imp_AR_gen: |
|
1591 fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set" |
|
1592 assumes "S retract_of T" "convex T" "T \<noteq> {}" "S homeomorphic S'" "closedin (top_of_set U) S'" |
|
1593 shows "S' retract_of U" |
|
1594 proof - |
|
1595 have "AR T" |
|
1596 by (simp add: assms convex_imp_AR) |
|
1597 then have "AR S" |
|
1598 using AR_retract_of_AR assms by auto |
|
1599 then show ?thesis |
|
1600 using assms AR_imp_absolute_retract by metis |
|
1601 qed |
|
1602 |
|
1603 lemma absolute_retract_imp_AR: |
|
1604 fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set" |
|
1605 assumes "S retract_of UNIV" "S homeomorphic S'" "closed S'" |
|
1606 shows "S' retract_of UNIV" |
|
1607 using AR_imp_absolute_retract_UNIV assms retract_of_UNIV by blast |
|
1608 |
|
1609 lemma homeomorphic_compact_arness: |
|
1610 fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set" |
|
1611 assumes "S homeomorphic S'" |
|
1612 shows "compact S \<and> S retract_of UNIV \<longleftrightarrow> compact S' \<and> S' retract_of UNIV" |
|
1613 using assms homeomorphic_compactness |
|
1614 apply auto |
|
1615 apply (meson assms compact_AR homeomorphic_AR_iff_AR homeomorphic_compactness)+ |
|
1616 done |
|
1617 |
|
1618 lemma absolute_retract_from_Un_Int: |
|
1619 fixes S :: "'a::euclidean_space set" |
|
1620 assumes "(S \<union> T) retract_of UNIV" "(S \<inter> T) retract_of UNIV" "closed S" "closed T" |
|
1621 shows "S retract_of UNIV" |
|
1622 using AR_from_Un_Int assms retract_of_UNIV by auto |
|
1623 |
|
1624 lemma ENR_from_Un_Int_gen: |
|
1625 fixes S :: "'a::euclidean_space set" |
|
1626 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)" |
|
1627 shows "ENR S" |
|
1628 apply (simp add: ENR_ANR) |
|
1629 using ANR_from_Un_Int_local ENR_ANR assms locally_compact_closedin by blast |
|
1630 |
|
1631 |
|
1632 lemma ENR_from_Un_Int: |
|
1633 fixes S :: "'a::euclidean_space set" |
|
1634 assumes "closed S" "closed T" "ENR(S \<union> T)" "ENR(S \<inter> T)" |
|
1635 shows "ENR S" |
|
1636 by (meson ENR_from_Un_Int_gen assms closed_subset sup_ge1 sup_ge2) |
|
1637 |
|
1638 |
|
1639 lemma ENR_finite_Union_convex_closed: |
|
1640 fixes \<T> :: "'a::euclidean_space set set" |
|
1641 assumes \<T>: "finite \<T>" and clo: "\<And>C. C \<in> \<T> \<Longrightarrow> closed C" and con: "\<And>C. C \<in> \<T> \<Longrightarrow> convex C" |
|
1642 shows "ENR(\<Union> \<T>)" |
|
1643 by (simp add: ENR_ANR ANR_finite_Union_convex_closed \<T> clo closed_Union closed_imp_locally_compact con) |
|
1644 |
|
1645 lemma finite_imp_ENR: |
|
1646 fixes S :: "'a::euclidean_space set" |
|
1647 shows "finite S \<Longrightarrow> ENR S" |
|
1648 by (simp add: ENR_ANR finite_imp_ANR finite_imp_closed closed_imp_locally_compact) |
|
1649 |
|
1650 lemma ENR_insert: |
|
1651 fixes S :: "'a::euclidean_space set" |
|
1652 assumes "closed S" "ENR S" |
|
1653 shows "ENR(insert a S)" |
|
1654 proof - |
|
1655 have "ENR ({a} \<union> S)" |
|
1656 by (metis ANR_insert ENR_ANR Un_commute Un_insert_right assms closed_imp_locally_compact closed_insert sup_bot_right) |
|
1657 then show ?thesis |
|
1658 by auto |
|
1659 qed |
|
1660 |
|
1661 lemma ENR_path_component_ENR: |
|
1662 fixes S :: "'a::euclidean_space set" |
|
1663 assumes "ENR S" |
|
1664 shows "ENR(path_component_set S x)" |
|
1665 by (metis ANR_imp_locally_path_connected ENR_empty ENR_imp_ANR ENR_openin assms |
|
1666 locally_path_connected_2 openin_subtopology_self path_component_eq_empty) |
|
1667 |
|
1668 (*UNUSED |
|
1669 lemma ENR_Times: |
|
1670 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
|
1671 assumes "ENR S" "ENR T" shows "ENR(S \<times> T)" |
|
1672 using assms apply (simp add: ENR_ANR ANR_Times) |
|
1673 thm locally_compact_Times |
|
1674 oops |
|
1675 SIMP_TAC[ENR_ANR; ANR_PCROSS; LOCALLY_COMPACT_PCROSS]);; |
|
1676 *) |
|
1677 |
|
1678 subsection\<open>Finally, spheres are ANRs and ENRs\<close> |
|
1679 |
|
1680 lemma absolute_retract_homeomorphic_convex_compact: |
|
1681 fixes S :: "'a::euclidean_space set" and U :: "'b::euclidean_space set" |
|
1682 assumes "S homeomorphic U" "S \<noteq> {}" "S \<subseteq> T" "convex U" "compact U" |
|
1683 shows "S retract_of T" |
|
1684 by (metis UNIV_I assms compact_AR convex_imp_AR homeomorphic_AR_iff_AR homeomorphic_compactness homeomorphic_empty(1) retract_of_subset subsetI) |
|
1685 |
|
1686 lemma frontier_retract_of_punctured_universe: |
|
1687 fixes S :: "'a::euclidean_space set" |
|
1688 assumes "convex S" "bounded S" "a \<in> interior S" |
|
1689 shows "(frontier S) retract_of (- {a})" |
|
1690 using rel_frontier_retract_of_punctured_affine_hull |
|
1691 by (metis Compl_eq_Diff_UNIV affine_hull_nonempty_interior assms empty_iff rel_frontier_frontier rel_interior_nonempty_interior) |
|
1692 |
|
1693 lemma sphere_retract_of_punctured_universe_gen: |
|
1694 fixes a :: "'a::euclidean_space" |
|
1695 assumes "b \<in> ball a r" |
|
1696 shows "sphere a r retract_of (- {b})" |
|
1697 proof - |
|
1698 have "frontier (cball a r) retract_of (- {b})" |
|
1699 apply (rule frontier_retract_of_punctured_universe) |
|
1700 using assms by auto |
|
1701 then show ?thesis |
|
1702 by simp |
|
1703 qed |
|
1704 |
|
1705 lemma sphere_retract_of_punctured_universe: |
|
1706 fixes a :: "'a::euclidean_space" |
|
1707 assumes "0 < r" |
|
1708 shows "sphere a r retract_of (- {a})" |
|
1709 by (simp add: assms sphere_retract_of_punctured_universe_gen) |
|
1710 |
|
1711 lemma ENR_sphere: |
|
1712 fixes a :: "'a::euclidean_space" |
|
1713 shows "ENR(sphere a r)" |
|
1714 proof (cases "0 < r") |
|
1715 case True |
|
1716 then have "sphere a r retract_of -{a}" |
|
1717 by (simp add: sphere_retract_of_punctured_universe) |
|
1718 with open_delete show ?thesis |
|
1719 by (auto simp: ENR_def) |
|
1720 next |
|
1721 case False |
|
1722 then show ?thesis |
|
1723 using finite_imp_ENR |
|
1724 by (metis finite_insert infinite_imp_nonempty less_linear sphere_eq_empty sphere_trivial) |
|
1725 qed |
|
1726 |
|
1727 corollary\<^marker>\<open>tag unimportant\<close> ANR_sphere: |
|
1728 fixes a :: "'a::euclidean_space" |
|
1729 shows "ANR(sphere a r)" |
|
1730 by (simp add: ENR_imp_ANR ENR_sphere) |
|
1731 |
|
1732 subsection\<open>Spheres are connected, etc\<close> |
|
1733 |
|
1734 lemma locally_path_connected_sphere_gen: |
|
1735 fixes S :: "'a::euclidean_space set" |
|
1736 assumes "bounded S" and "convex S" |
|
1737 shows "locally path_connected (rel_frontier S)" |
|
1738 proof (cases "rel_interior S = {}") |
|
1739 case True |
|
1740 with assms show ?thesis |
|
1741 by (simp add: rel_interior_eq_empty) |
|
1742 next |
|
1743 case False |
|
1744 then obtain a where a: "a \<in> rel_interior S" |
|
1745 by blast |
|
1746 show ?thesis |
|
1747 proof (rule retract_of_locally_path_connected) |
|
1748 show "locally path_connected (affine hull S - {a})" |
|
1749 by (meson convex_affine_hull convex_imp_locally_path_connected locally_open_subset openin_delete openin_subtopology_self) |
|
1750 show "rel_frontier S retract_of affine hull S - {a}" |
|
1751 using a assms rel_frontier_retract_of_punctured_affine_hull by blast |
|
1752 qed |
|
1753 qed |
|
1754 |
|
1755 lemma locally_connected_sphere_gen: |
|
1756 fixes S :: "'a::euclidean_space set" |
|
1757 assumes "bounded S" and "convex S" |
|
1758 shows "locally connected (rel_frontier S)" |
|
1759 by (simp add: ANR_imp_locally_connected ANR_rel_frontier_convex assms) |
|
1760 |
|
1761 lemma locally_path_connected_sphere: |
|
1762 fixes a :: "'a::euclidean_space" |
|
1763 shows "locally path_connected (sphere a r)" |
|
1764 using ENR_imp_locally_path_connected ENR_sphere by blast |
|
1765 |
|
1766 lemma locally_connected_sphere: |
|
1767 fixes a :: "'a::euclidean_space" |
|
1768 shows "locally connected(sphere a r)" |
|
1769 using ANR_imp_locally_connected ANR_sphere by blast |
|
1770 |
|
1771 subsection\<open>Borsuk homotopy extension theorem\<close> |
|
1772 |
|
1773 text\<open>It's only this late so we can use the concept of retraction, |
|
1774 saying that the domain sets or range set are ENRs.\<close> |
|
1775 |
|
1776 theorem Borsuk_homotopy_extension_homotopic: |
|
1777 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
|
1778 assumes cloTS: "closedin (top_of_set T) S" |
|
1779 and anr: "(ANR S \<and> ANR T) \<or> ANR U" |
|
1780 and contf: "continuous_on T f" |
|
1781 and "f ` T \<subseteq> U" |
|
1782 and "homotopic_with_canon (\<lambda>x. True) S U f g" |
|
1783 obtains g' where "homotopic_with_canon (\<lambda>x. True) T U f g'" |
|
1784 "continuous_on T g'" "image g' T \<subseteq> U" |
|
1785 "\<And>x. x \<in> S \<Longrightarrow> g' x = g x" |
|
1786 proof - |
|
1787 have "S \<subseteq> T" using assms closedin_imp_subset by blast |
|
1788 obtain h where conth: "continuous_on ({0..1} \<times> S) h" |
|
1789 and him: "h ` ({0..1} \<times> S) \<subseteq> U" |
|
1790 and [simp]: "\<And>x. h(0, x) = f x" "\<And>x. h(1::real, x) = g x" |
|
1791 using assms by (auto simp: homotopic_with_def) |
|
1792 define h' where "h' \<equiv> \<lambda>z. if snd z \<in> S then h z else (f \<circ> snd) z" |
|
1793 define B where "B \<equiv> {0::real} \<times> T \<union> {0..1} \<times> S" |
|
1794 have clo0T: "closedin (top_of_set ({0..1} \<times> T)) ({0::real} \<times> T)" |
|
1795 by (simp add: Abstract_Topology.closedin_Times) |
|
1796 moreover have cloT1S: "closedin (top_of_set ({0..1} \<times> T)) ({0..1} \<times> S)" |
|
1797 by (simp add: Abstract_Topology.closedin_Times assms) |
|
1798 ultimately have clo0TB:"closedin (top_of_set ({0..1} \<times> T)) B" |
|
1799 by (auto simp: B_def) |
|
1800 have cloBS: "closedin (top_of_set B) ({0..1} \<times> S)" |
|
1801 by (metis (no_types) Un_subset_iff B_def closedin_subset_trans [OF cloT1S] clo0TB closedin_imp_subset closedin_self) |
|
1802 moreover have cloBT: "closedin (top_of_set B) ({0} \<times> T)" |
|
1803 using \<open>S \<subseteq> T\<close> closedin_subset_trans [OF clo0T] |
|
1804 by (metis B_def Un_upper1 clo0TB closedin_closed inf_le1) |
|
1805 moreover have "continuous_on ({0} \<times> T) (f \<circ> snd)" |
|
1806 apply (rule continuous_intros)+ |
|
1807 apply (simp add: contf) |
|
1808 done |
|
1809 ultimately have conth': "continuous_on B h'" |
|
1810 apply (simp add: h'_def B_def Un_commute [of "{0} \<times> T"]) |
|
1811 apply (auto intro!: continuous_on_cases_local conth) |
|
1812 done |
|
1813 have "image h' B \<subseteq> U" |
|
1814 using \<open>f ` T \<subseteq> U\<close> him by (auto simp: h'_def B_def) |
|
1815 obtain V k where "B \<subseteq> V" and opeTV: "openin (top_of_set ({0..1} \<times> T)) V" |
|
1816 and contk: "continuous_on V k" and kim: "k ` V \<subseteq> U" |
|
1817 and keq: "\<And>x. x \<in> B \<Longrightarrow> k x = h' x" |
|
1818 using anr |
|
1819 proof |
|
1820 assume ST: "ANR S \<and> ANR T" |
|
1821 have eq: "({0} \<times> T \<inter> {0..1} \<times> S) = {0::real} \<times> S" |
|
1822 using \<open>S \<subseteq> T\<close> by auto |
|
1823 have "ANR B" |
|
1824 apply (simp add: B_def) |
|
1825 apply (rule ANR_closed_Un_local) |
|
1826 apply (metis cloBT B_def) |
|
1827 apply (metis Un_commute cloBS B_def) |
|
1828 apply (simp_all add: ANR_Times convex_imp_ANR ANR_singleton ST eq) |
|
1829 done |
|
1830 note Vk = that |
|
1831 have *: thesis if "openin (top_of_set ({0..1::real} \<times> T)) V" |
|
1832 "retraction V B r" for V r |
|
1833 using that |
|
1834 apply (clarsimp simp add: retraction_def) |
|
1835 apply (rule Vk [of V "h' \<circ> r"], assumption+) |
|
1836 apply (metis continuous_on_compose conth' continuous_on_subset) |
|
1837 using \<open>h' ` B \<subseteq> U\<close> apply force+ |
|
1838 done |
|
1839 show thesis |
|
1840 apply (rule ANR_imp_neighbourhood_retract [OF \<open>ANR B\<close> clo0TB]) |
|
1841 apply (auto simp: ANR_Times ANR_singleton ST retract_of_def *) |
|
1842 done |
|
1843 next |
|
1844 assume "ANR U" |
|
1845 with ANR_imp_absolute_neighbourhood_extensor \<open>h' ` B \<subseteq> U\<close> clo0TB conth' that |
|
1846 show ?thesis by blast |
|
1847 qed |
|
1848 define S' where "S' \<equiv> {x. \<exists>u::real. u \<in> {0..1} \<and> (u, x::'a) \<in> {0..1} \<times> T - V}" |
|
1849 have "closedin (top_of_set T) S'" |
|
1850 unfolding S'_def |
|
1851 apply (rule closedin_compact_projection, blast) |
|
1852 using closedin_self opeTV by blast |
|
1853 have S'_def: "S' = {x. \<exists>u::real. (u, x::'a) \<in> {0..1} \<times> T - V}" |
|
1854 by (auto simp: S'_def) |
|
1855 have cloTS': "closedin (top_of_set T) S'" |
|
1856 using S'_def \<open>closedin (top_of_set T) S'\<close> by blast |
|
1857 have "S \<inter> S' = {}" |
|
1858 using S'_def B_def \<open>B \<subseteq> V\<close> by force |
|
1859 obtain a :: "'a \<Rightarrow> real" where conta: "continuous_on T a" |
|
1860 and "\<And>x. x \<in> T \<Longrightarrow> a x \<in> closed_segment 1 0" |
|
1861 and a1: "\<And>x. x \<in> S \<Longrightarrow> a x = 1" |
|
1862 and a0: "\<And>x. x \<in> S' \<Longrightarrow> a x = 0" |
|
1863 apply (rule Urysohn_local [OF cloTS cloTS' \<open>S \<inter> S' = {}\<close>, of 1 0], blast) |
|
1864 done |
|
1865 then have ain: "\<And>x. x \<in> T \<Longrightarrow> a x \<in> {0..1}" |
|
1866 using closed_segment_eq_real_ivl by auto |
|
1867 have inV: "(u * a t, t) \<in> V" if "t \<in> T" "0 \<le> u" "u \<le> 1" for t u |
|
1868 proof (rule ccontr) |
|
1869 assume "(u * a t, t) \<notin> V" |
|
1870 with ain [OF \<open>t \<in> T\<close>] have "a t = 0" |
|
1871 apply simp |
|
1872 apply (rule a0) |
|
1873 by (metis (no_types, lifting) Diff_iff S'_def SigmaI atLeastAtMost_iff mem_Collect_eq mult_le_one mult_nonneg_nonneg that) |
|
1874 show False |
|
1875 using B_def \<open>(u * a t, t) \<notin> V\<close> \<open>B \<subseteq> V\<close> \<open>a t = 0\<close> that by auto |
|
1876 qed |
|
1877 show ?thesis |
|
1878 proof |
|
1879 show hom: "homotopic_with_canon (\<lambda>x. True) T U f (\<lambda>x. k (a x, x))" |
|
1880 proof (simp add: homotopic_with, intro exI conjI) |
|
1881 show "continuous_on ({0..1} \<times> T) (k \<circ> (\<lambda>z. (fst z *\<^sub>R (a \<circ> snd) z, snd z)))" |
|
1882 apply (intro continuous_on_compose continuous_intros) |
|
1883 apply (rule continuous_on_subset [OF conta], force) |
|
1884 apply (rule continuous_on_subset [OF contk]) |
|
1885 apply (force intro: inV) |
|
1886 done |
|
1887 show "(k \<circ> (\<lambda>z. (fst z *\<^sub>R (a \<circ> snd) z, snd z))) ` ({0..1} \<times> T) \<subseteq> U" |
|
1888 using inV kim by auto |
|
1889 show "\<forall>x\<in>T. (k \<circ> (\<lambda>z. (fst z *\<^sub>R (a \<circ> snd) z, snd z))) (0, x) = f x" |
|
1890 by (simp add: B_def h'_def keq) |
|
1891 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)" |
|
1892 by auto |
|
1893 qed |
|
1894 show "continuous_on T (\<lambda>x. k (a x, x))" |
|
1895 using homotopic_with_imp_continuous_maps [OF hom] by auto |
|
1896 show "(\<lambda>x. k (a x, x)) ` T \<subseteq> U" |
|
1897 proof clarify |
|
1898 fix t |
|
1899 assume "t \<in> T" |
|
1900 show "k (a t, t) \<in> U" |
|
1901 by (metis \<open>t \<in> T\<close> image_subset_iff inV kim not_one_le_zero linear mult_cancel_right1) |
|
1902 qed |
|
1903 show "\<And>x. x \<in> S \<Longrightarrow> k (a x, x) = g x" |
|
1904 by (simp add: B_def a1 h'_def keq) |
|
1905 qed |
|
1906 qed |
|
1907 |
|
1908 |
|
1909 corollary\<^marker>\<open>tag unimportant\<close> nullhomotopic_into_ANR_extension: |
|
1910 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
|
1911 assumes "closed S" |
|
1912 and contf: "continuous_on S f" |
|
1913 and "ANR T" |
|
1914 and fim: "f ` S \<subseteq> T" |
|
1915 and "S \<noteq> {}" |
|
1916 shows "(\<exists>c. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c)) \<longleftrightarrow> |
|
1917 (\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> T \<and> (\<forall>x \<in> S. g x = f x))" |
|
1918 (is "?lhs = ?rhs") |
|
1919 proof |
|
1920 assume ?lhs |
|
1921 then obtain c where c: "homotopic_with_canon (\<lambda>x. True) S T (\<lambda>x. c) f" |
|
1922 by (blast intro: homotopic_with_symD) |
|
1923 have "closedin (top_of_set UNIV) S" |
|
1924 using \<open>closed S\<close> closed_closedin by fastforce |
|
1925 then obtain g where "continuous_on UNIV g" "range g \<subseteq> T" |
|
1926 "\<And>x. x \<in> S \<Longrightarrow> g x = f x" |
|
1927 apply (rule Borsuk_homotopy_extension_homotopic [OF _ _ continuous_on_const _ c, where T=UNIV]) |
|
1928 using \<open>ANR T\<close> \<open>S \<noteq> {}\<close> c homotopic_with_imp_subset1 apply fastforce+ |
|
1929 done |
|
1930 then show ?rhs by blast |
|
1931 next |
|
1932 assume ?rhs |
|
1933 then obtain g where "continuous_on UNIV g" "range g \<subseteq> T" "\<And>x. x\<in>S \<Longrightarrow> g x = f x" |
|
1934 by blast |
|
1935 then obtain c where "homotopic_with_canon (\<lambda>h. True) UNIV T g (\<lambda>x. c)" |
|
1936 using nullhomotopic_from_contractible [of UNIV g T] contractible_UNIV by blast |
|
1937 then have "homotopic_with_canon (\<lambda>x. True) S T g (\<lambda>x. c)" |
|
1938 by (simp add: homotopic_from_subtopology) |
|
1939 then show ?lhs |
|
1940 by (force elim: homotopic_with_eq [of _ _ _ g "\<lambda>x. c"] simp: \<open>\<And>x. x \<in> S \<Longrightarrow> g x = f x\<close>) |
|
1941 qed |
|
1942 |
|
1943 corollary\<^marker>\<open>tag unimportant\<close> nullhomotopic_into_rel_frontier_extension: |
|
1944 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
|
1945 assumes "closed S" |
|
1946 and contf: "continuous_on S f" |
|
1947 and "convex T" "bounded T" |
|
1948 and fim: "f ` S \<subseteq> rel_frontier T" |
|
1949 and "S \<noteq> {}" |
|
1950 shows "(\<exists>c. homotopic_with_canon (\<lambda>x. True) S (rel_frontier T) f (\<lambda>x. c)) \<longleftrightarrow> |
|
1951 (\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> rel_frontier T \<and> (\<forall>x \<in> S. g x = f x))" |
|
1952 by (simp add: nullhomotopic_into_ANR_extension assms ANR_rel_frontier_convex) |
|
1953 |
|
1954 corollary\<^marker>\<open>tag unimportant\<close> nullhomotopic_into_sphere_extension: |
|
1955 fixes f :: "'a::euclidean_space \<Rightarrow> 'b :: euclidean_space" |
|
1956 assumes "closed S" and contf: "continuous_on S f" |
|
1957 and "S \<noteq> {}" and fim: "f ` S \<subseteq> sphere a r" |
|
1958 shows "((\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere a r) f (\<lambda>x. c)) \<longleftrightarrow> |
|
1959 (\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> sphere a r \<and> (\<forall>x \<in> S. g x = f x)))" |
|
1960 (is "?lhs = ?rhs") |
|
1961 proof (cases "r = 0") |
|
1962 case True with fim show ?thesis |
|
1963 apply auto |
|
1964 using fim continuous_on_const apply fastforce |
|
1965 by (metis contf contractible_sing nullhomotopic_into_contractible) |
|
1966 next |
|
1967 case False |
|
1968 then have eq: "sphere a r = rel_frontier (cball a r)" by simp |
|
1969 show ?thesis |
|
1970 using fim unfolding eq |
|
1971 apply (rule nullhomotopic_into_rel_frontier_extension [OF \<open>closed S\<close> contf convex_cball bounded_cball]) |
|
1972 apply (rule \<open>S \<noteq> {}\<close>) |
|
1973 done |
|
1974 qed |
|
1975 |
|
1976 proposition\<^marker>\<open>tag unimportant\<close> Borsuk_map_essential_bounded_component: |
|
1977 fixes a :: "'a :: euclidean_space" |
|
1978 assumes "compact S" and "a \<notin> S" |
|
1979 shows "bounded (connected_component_set (- S) a) \<longleftrightarrow> |
|
1980 \<not>(\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) |
|
1981 (\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a)) (\<lambda>x. c))" |
|
1982 (is "?lhs = ?rhs") |
|
1983 proof (cases "S = {}") |
|
1984 case True then show ?thesis |
|
1985 by simp |
|
1986 next |
|
1987 case False |
|
1988 have "closed S" "bounded S" |
|
1989 using \<open>compact S\<close> compact_eq_bounded_closed by auto |
|
1990 have s01: "(\<lambda>x. (x - a) /\<^sub>R norm (x - a)) ` S \<subseteq> sphere 0 1" |
|
1991 using \<open>a \<notin> S\<close> by clarsimp (metis dist_eq_0_iff dist_norm mult.commute right_inverse) |
|
1992 have aincc: "a \<in> connected_component_set (- S) a" |
|
1993 by (simp add: \<open>a \<notin> S\<close>) |
|
1994 obtain r where "r>0" and r: "S \<subseteq> ball 0 r" |
|
1995 using bounded_subset_ballD \<open>bounded S\<close> by blast |
|
1996 have "\<not> ?rhs \<longleftrightarrow> \<not> ?lhs" |
|
1997 proof |
|
1998 assume notr: "\<not> ?rhs" |
|
1999 have nog: "\<nexists>g. continuous_on (S \<union> connected_component_set (- S) a) g \<and> |
|
2000 g ` (S \<union> connected_component_set (- S) a) \<subseteq> sphere 0 1 \<and> |
|
2001 (\<forall>x\<in>S. g x = (x - a) /\<^sub>R norm (x - a))" |
|
2002 if "bounded (connected_component_set (- S) a)" |
|
2003 apply (rule non_extensible_Borsuk_map [OF \<open>compact S\<close> componentsI _ aincc]) |
|
2004 using \<open>a \<notin> S\<close> that by auto |
|
2005 obtain g where "range g \<subseteq> sphere 0 1" "continuous_on UNIV g" |
|
2006 "\<And>x. x \<in> S \<Longrightarrow> g x = (x - a) /\<^sub>R norm (x - a)" |
|
2007 using notr |
|
2008 by (auto simp: nullhomotopic_into_sphere_extension |
|
2009 [OF \<open>closed S\<close> continuous_on_Borsuk_map [OF \<open>a \<notin> S\<close>] False s01]) |
|
2010 with \<open>a \<notin> S\<close> show "\<not> ?lhs" |
|
2011 apply (clarsimp simp: Borsuk_map_into_sphere [of a S, symmetric] dest!: nog) |
|
2012 apply (drule_tac x=g in spec) |
|
2013 using continuous_on_subset by fastforce |
|
2014 next |
|
2015 assume "\<not> ?lhs" |
|
2016 then obtain b where b: "b \<in> connected_component_set (- S) a" and "r \<le> norm b" |
|
2017 using bounded_iff linear by blast |
|
2018 then have bnot: "b \<notin> ball 0 r" |
|
2019 by simp |
|
2020 have "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) (\<lambda>x. (x - a) /\<^sub>R norm (x - a)) |
|
2021 (\<lambda>x. (x - b) /\<^sub>R norm (x - b))" |
|
2022 apply (rule Borsuk_maps_homotopic_in_path_component) |
|
2023 using \<open>closed S\<close> b open_Compl open_path_connected_component apply fastforce |
|
2024 done |
|
2025 moreover |
|
2026 obtain c where "homotopic_with_canon (\<lambda>x. True) (ball 0 r) (sphere 0 1) |
|
2027 (\<lambda>x. inverse (norm (x - b)) *\<^sub>R (x - b)) (\<lambda>x. c)" |
|
2028 proof (rule nullhomotopic_from_contractible) |
|
2029 show "contractible (ball (0::'a) r)" |
|
2030 by (metis convex_imp_contractible convex_ball) |
|
2031 show "continuous_on (ball 0 r) (\<lambda>x. inverse(norm (x - b)) *\<^sub>R (x - b))" |
|
2032 by (rule continuous_on_Borsuk_map [OF bnot]) |
|
2033 show "(\<lambda>x. (x - b) /\<^sub>R norm (x - b)) ` ball 0 r \<subseteq> sphere 0 1" |
|
2034 using bnot Borsuk_map_into_sphere by blast |
|
2035 qed blast |
|
2036 ultimately have "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) (\<lambda>x. (x - a) /\<^sub>R norm (x - a)) (\<lambda>x. c)" |
|
2037 by (meson homotopic_with_subset_left homotopic_with_trans r) |
|
2038 then show "\<not> ?rhs" |
|
2039 by blast |
|
2040 qed |
|
2041 then show ?thesis by blast |
|
2042 qed |
|
2043 |
|
2044 lemma homotopic_Borsuk_maps_in_bounded_component: |
|
2045 fixes a :: "'a :: euclidean_space" |
|
2046 assumes "compact S" and "a \<notin> S"and "b \<notin> S" |
|
2047 and boc: "bounded (connected_component_set (- S) a)" |
|
2048 and hom: "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) |
|
2049 (\<lambda>x. (x - a) /\<^sub>R norm (x - a)) |
|
2050 (\<lambda>x. (x - b) /\<^sub>R norm (x - b))" |
|
2051 shows "connected_component (- S) a b" |
|
2052 proof (rule ccontr) |
|
2053 assume notcc: "\<not> connected_component (- S) a b" |
|
2054 let ?T = "S \<union> connected_component_set (- S) a" |
|
2055 have "\<nexists>g. continuous_on (S \<union> connected_component_set (- S) a) g \<and> |
|
2056 g ` (S \<union> connected_component_set (- S) a) \<subseteq> sphere 0 1 \<and> |
|
2057 (\<forall>x\<in>S. g x = (x - a) /\<^sub>R norm (x - a))" |
|
2058 by (simp add: \<open>a \<notin> S\<close> componentsI non_extensible_Borsuk_map [OF \<open>compact S\<close> _ boc]) |
|
2059 moreover obtain g where "continuous_on (S \<union> connected_component_set (- S) a) g" |
|
2060 "g ` (S \<union> connected_component_set (- S) a) \<subseteq> sphere 0 1" |
|
2061 "\<And>x. x \<in> S \<Longrightarrow> g x = (x - a) /\<^sub>R norm (x - a)" |
|
2062 proof (rule Borsuk_homotopy_extension_homotopic) |
|
2063 show "closedin (top_of_set ?T) S" |
|
2064 by (simp add: \<open>compact S\<close> closed_subset compact_imp_closed) |
|
2065 show "continuous_on ?T (\<lambda>x. (x - b) /\<^sub>R norm (x - b))" |
|
2066 by (simp add: \<open>b \<notin> S\<close> notcc continuous_on_Borsuk_map) |
|
2067 show "(\<lambda>x. (x - b) /\<^sub>R norm (x - b)) ` ?T \<subseteq> sphere 0 1" |
|
2068 by (simp add: \<open>b \<notin> S\<close> notcc Borsuk_map_into_sphere) |
|
2069 show "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) |
|
2070 (\<lambda>x. (x - b) /\<^sub>R norm (x - b)) (\<lambda>x. (x - a) /\<^sub>R norm (x - a))" |
|
2071 by (simp add: hom homotopic_with_symD) |
|
2072 qed (auto simp: ANR_sphere intro: that) |
|
2073 ultimately show False by blast |
|
2074 qed |
|
2075 |
|
2076 |
|
2077 lemma Borsuk_maps_homotopic_in_connected_component_eq: |
|
2078 fixes a :: "'a :: euclidean_space" |
|
2079 assumes S: "compact S" "a \<notin> S" "b \<notin> S" and 2: "2 \<le> DIM('a)" |
|
2080 shows "(homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) |
|
2081 (\<lambda>x. (x - a) /\<^sub>R norm (x - a)) |
|
2082 (\<lambda>x. (x - b) /\<^sub>R norm (x - b)) \<longleftrightarrow> |
|
2083 connected_component (- S) a b)" |
|
2084 (is "?lhs = ?rhs") |
|
2085 proof |
|
2086 assume L: ?lhs |
|
2087 show ?rhs |
|
2088 proof (cases "bounded(connected_component_set (- S) a)") |
|
2089 case True |
|
2090 show ?thesis |
|
2091 by (rule homotopic_Borsuk_maps_in_bounded_component [OF S True L]) |
|
2092 next |
|
2093 case not_bo_a: False |
|
2094 show ?thesis |
|
2095 proof (cases "bounded(connected_component_set (- S) b)") |
|
2096 case True |
|
2097 show ?thesis |
|
2098 using homotopic_Borsuk_maps_in_bounded_component [OF S] |
|
2099 by (simp add: L True assms connected_component_sym homotopic_Borsuk_maps_in_bounded_component homotopic_with_sym) |
|
2100 next |
|
2101 case False |
|
2102 then show ?thesis |
|
2103 using cobounded_unique_unbounded_component [of "-S" a b] \<open>compact S\<close> not_bo_a |
|
2104 by (auto simp: compact_eq_bounded_closed assms connected_component_eq_eq) |
|
2105 qed |
|
2106 qed |
|
2107 next |
|
2108 assume R: ?rhs |
|
2109 then have "path_component (- S) a b" |
|
2110 using assms(1) compact_eq_bounded_closed open_Compl open_path_connected_component_set by fastforce |
|
2111 then show ?lhs |
|
2112 by (simp add: Borsuk_maps_homotopic_in_path_component) |
|
2113 qed |
|
2114 |
|
2115 subsection\<open>More extension theorems\<close> |
|
2116 |
|
2117 lemma extension_from_clopen: |
|
2118 assumes ope: "openin (top_of_set S) T" |
|
2119 and clo: "closedin (top_of_set S) T" |
|
2120 and contf: "continuous_on T f" and fim: "f ` T \<subseteq> U" and null: "U = {} \<Longrightarrow> S = {}" |
|
2121 obtains g where "continuous_on S g" "g ` S \<subseteq> U" "\<And>x. x \<in> T \<Longrightarrow> g x = f x" |
|
2122 proof (cases "U = {}") |
|
2123 case True |
|
2124 then show ?thesis |
|
2125 by (simp add: null that) |
|
2126 next |
|
2127 case False |
|
2128 then obtain a where "a \<in> U" |
|
2129 by auto |
|
2130 let ?g = "\<lambda>x. if x \<in> T then f x else a" |
|
2131 have Seq: "S = T \<union> (S - T)" |
|
2132 using clo closedin_imp_subset by fastforce |
|
2133 show ?thesis |
|
2134 proof |
|
2135 have "continuous_on (T \<union> (S - T)) ?g" |
|
2136 apply (rule continuous_on_cases_local) |
|
2137 using Seq clo ope by (auto simp: contf continuous_on_const intro: continuous_on_cases_local) |
|
2138 with Seq show "continuous_on S ?g" |
|
2139 by metis |
|
2140 show "?g ` S \<subseteq> U" |
|
2141 using \<open>a \<in> U\<close> fim by auto |
|
2142 show "\<And>x. x \<in> T \<Longrightarrow> ?g x = f x" |
|
2143 by auto |
|
2144 qed |
|
2145 qed |
|
2146 |
|
2147 |
|
2148 lemma extension_from_component: |
|
2149 fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space" |
|
2150 assumes S: "locally connected S \<or> compact S" and "ANR U" |
|
2151 and C: "C \<in> components S" and contf: "continuous_on C f" and fim: "f ` C \<subseteq> U" |
|
2152 obtains g where "continuous_on S g" "g ` S \<subseteq> U" "\<And>x. x \<in> C \<Longrightarrow> g x = f x" |
|
2153 proof - |
|
2154 obtain T g where ope: "openin (top_of_set S) T" |
|
2155 and clo: "closedin (top_of_set S) T" |
|
2156 and "C \<subseteq> T" and contg: "continuous_on T g" and gim: "g ` T \<subseteq> U" |
|
2157 and gf: "\<And>x. x \<in> C \<Longrightarrow> g x = f x" |
|
2158 using S |
|
2159 proof |
|
2160 assume "locally connected S" |
|
2161 show ?thesis |
|
2162 by (metis C \<open>locally connected S\<close> openin_components_locally_connected closedin_component contf fim order_refl that) |
|
2163 next |
|
2164 assume "compact S" |
|
2165 then obtain W g where "C \<subseteq> W" and opeW: "openin (top_of_set S) W" |
|
2166 and contg: "continuous_on W g" |
|
2167 and gim: "g ` W \<subseteq> U" and gf: "\<And>x. x \<in> C \<Longrightarrow> g x = f x" |
|
2168 using ANR_imp_absolute_neighbourhood_extensor [of U C f S] C \<open>ANR U\<close> closedin_component contf fim by blast |
|
2169 then obtain V where "open V" and V: "W = S \<inter> V" |
|
2170 by (auto simp: openin_open) |
|
2171 moreover have "locally compact S" |
|
2172 by (simp add: \<open>compact S\<close> closed_imp_locally_compact compact_imp_closed) |
|
2173 ultimately obtain K where opeK: "openin (top_of_set S) K" and "compact K" "C \<subseteq> K" "K \<subseteq> V" |
|
2174 by (metis C Int_subset_iff \<open>C \<subseteq> W\<close> \<open>compact S\<close> compact_components Sura_Bura_clopen_subset) |
|
2175 show ?thesis |
|
2176 proof |
|
2177 show "closedin (top_of_set S) K" |
|
2178 by (meson \<open>compact K\<close> \<open>compact S\<close> closedin_compact_eq opeK openin_imp_subset) |
|
2179 show "continuous_on K g" |
|
2180 by (metis Int_subset_iff V \<open>K \<subseteq> V\<close> contg continuous_on_subset opeK openin_subtopology subset_eq) |
|
2181 show "g ` K \<subseteq> U" |
|
2182 using V \<open>K \<subseteq> V\<close> gim opeK openin_imp_subset by fastforce |
|
2183 qed (use opeK gf \<open>C \<subseteq> K\<close> in auto) |
|
2184 qed |
|
2185 obtain h where "continuous_on S h" "h ` S \<subseteq> U" "\<And>x. x \<in> T \<Longrightarrow> h x = g x" |
|
2186 using extension_from_clopen |
|
2187 by (metis C bot.extremum_uniqueI clo contg gim fim image_is_empty in_components_nonempty ope) |
|
2188 then show ?thesis |
|
2189 by (metis \<open>C \<subseteq> T\<close> gf subset_eq that) |
|
2190 qed |
|
2191 |
|
2192 |
|
2193 lemma tube_lemma: |
|
2194 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
|
2195 assumes "compact S" and S: "S \<noteq> {}" "(\<lambda>x. (x,a)) ` S \<subseteq> U" |
|
2196 and ope: "openin (top_of_set (S \<times> T)) U" |
|
2197 obtains V where "openin (top_of_set T) V" "a \<in> V" "S \<times> V \<subseteq> U" |
|
2198 proof - |
|
2199 let ?W = "{y. \<exists>x. x \<in> S \<and> (x, y) \<in> (S \<times> T - U)}" |
|
2200 have "U \<subseteq> S \<times> T" "closedin (top_of_set (S \<times> T)) (S \<times> T - U)" |
|
2201 using ope by (auto simp: openin_closedin_eq) |
|
2202 then have "closedin (top_of_set T) ?W" |
|
2203 using \<open>compact S\<close> closedin_compact_projection by blast |
|
2204 moreover have "a \<in> T - ?W" |
|
2205 using \<open>U \<subseteq> S \<times> T\<close> S by auto |
|
2206 moreover have "S \<times> (T - ?W) \<subseteq> U" |
|
2207 by auto |
|
2208 ultimately show ?thesis |
|
2209 by (metis (no_types, lifting) Sigma_cong closedin_def that topspace_euclidean_subtopology) |
|
2210 qed |
|
2211 |
|
2212 lemma tube_lemma_gen: |
|
2213 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
|
2214 assumes "compact S" "S \<noteq> {}" "T \<subseteq> T'" "S \<times> T \<subseteq> U" |
|
2215 and ope: "openin (top_of_set (S \<times> T')) U" |
|
2216 obtains V where "openin (top_of_set T') V" "T \<subseteq> V" "S \<times> V \<subseteq> U" |
|
2217 proof - |
|
2218 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" |
|
2219 using assms by (auto intro: tube_lemma [OF \<open>compact S\<close>]) |
|
2220 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" |
|
2221 by metis |
|
2222 show ?thesis |
|
2223 proof |
|
2224 show "openin (top_of_set T') (\<Union>(F ` T))" |
|
2225 using F by blast |
|
2226 show "T \<subseteq> \<Union>(F ` T)" |
|
2227 using F by blast |
|
2228 show "S \<times> \<Union>(F ` T) \<subseteq> U" |
|
2229 using F by auto |
|
2230 qed |
|
2231 qed |
|
2232 |
|
2233 proposition\<^marker>\<open>tag unimportant\<close> homotopic_neighbourhood_extension: |
|
2234 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
|
2235 assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> U" |
|
2236 and contg: "continuous_on S g" and gim: "g ` S \<subseteq> U" |
|
2237 and clo: "closedin (top_of_set S) T" |
|
2238 and "ANR U" and hom: "homotopic_with_canon (\<lambda>x. True) T U f g" |
|
2239 obtains V where "T \<subseteq> V" "openin (top_of_set S) V" |
|
2240 "homotopic_with_canon (\<lambda>x. True) V U f g" |
|
2241 proof - |
|
2242 have "T \<subseteq> S" |
|
2243 using clo closedin_imp_subset by blast |
|
2244 obtain h where conth: "continuous_on ({0..1::real} \<times> T) h" |
|
2245 and him: "h ` ({0..1} \<times> T) \<subseteq> U" |
|
2246 and h0: "\<And>x. h(0, x) = f x" and h1: "\<And>x. h(1, x) = g x" |
|
2247 using hom by (auto simp: homotopic_with_def) |
|
2248 define h' where "h' \<equiv> \<lambda>z. if fst z \<in> {0} then f(snd z) |
|
2249 else if fst z \<in> {1} then g(snd z) |
|
2250 else h z" |
|
2251 let ?S0 = "{0::real} \<times> S" and ?S1 = "{1::real} \<times> S" |
|
2252 have "continuous_on(?S0 \<union> (?S1 \<union> {0..1} \<times> T)) h'" |
|
2253 unfolding h'_def |
|
2254 proof (intro continuous_on_cases_local) |
|
2255 show "closedin (top_of_set (?S0 \<union> (?S1 \<union> {0..1} \<times> T))) ?S0" |
|
2256 "closedin (top_of_set (?S1 \<union> {0..1} \<times> T)) ?S1" |
|
2257 using \<open>T \<subseteq> S\<close> by (force intro: closedin_Times closedin_subset_trans [of "{0..1} \<times> S"])+ |
|
2258 show "closedin (top_of_set (?S0 \<union> (?S1 \<union> {0..1} \<times> T))) (?S1 \<union> {0..1} \<times> T)" |
|
2259 "closedin (top_of_set (?S1 \<union> {0..1} \<times> T)) ({0..1} \<times> T)" |
|
2260 using \<open>T \<subseteq> S\<close> by (force intro: clo closedin_Times closedin_subset_trans [of "{0..1} \<times> S"])+ |
|
2261 show "continuous_on (?S0) (\<lambda>x. f (snd x))" |
|
2262 by (intro continuous_intros continuous_on_compose2 [OF contf]) auto |
|
2263 show "continuous_on (?S1) (\<lambda>x. g (snd x))" |
|
2264 by (intro continuous_intros continuous_on_compose2 [OF contg]) auto |
|
2265 qed (use h0 h1 conth in auto) |
|
2266 then have "continuous_on ({0,1} \<times> S \<union> ({0..1} \<times> T)) h'" |
|
2267 by (metis Sigma_Un_distrib1 Un_assoc insert_is_Un) |
|
2268 moreover have "h' ` ({0,1} \<times> S \<union> {0..1} \<times> T) \<subseteq> U" |
|
2269 using fim gim him \<open>T \<subseteq> S\<close> unfolding h'_def by force |
|
2270 moreover have "closedin (top_of_set ({0..1::real} \<times> S)) ({0,1} \<times> S \<union> {0..1::real} \<times> T)" |
|
2271 by (intro closedin_Times closedin_Un clo) (simp_all add: closed_subset) |
|
2272 ultimately |
|
2273 obtain W k where W: "({0,1} \<times> S) \<union> ({0..1} \<times> T) \<subseteq> W" |
|
2274 and opeW: "openin (top_of_set ({0..1} \<times> S)) W" |
|
2275 and contk: "continuous_on W k" |
|
2276 and kim: "k ` W \<subseteq> U" |
|
2277 and kh': "\<And>x. x \<in> ({0,1} \<times> S) \<union> ({0..1} \<times> T) \<Longrightarrow> k x = h' x" |
|
2278 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"]) |
|
2279 obtain T' where opeT': "openin (top_of_set S) T'" |
|
2280 and "T \<subseteq> T'" and TW: "{0..1} \<times> T' \<subseteq> W" |
|
2281 using tube_lemma_gen [of "{0..1::real}" T S W] W \<open>T \<subseteq> S\<close> opeW by auto |
|
2282 moreover have "homotopic_with_canon (\<lambda>x. True) T' U f g" |
|
2283 proof (simp add: homotopic_with, intro exI conjI) |
|
2284 show "continuous_on ({0..1} \<times> T') k" |
|
2285 using TW continuous_on_subset contk by auto |
|
2286 show "k ` ({0..1} \<times> T') \<subseteq> U" |
|
2287 using TW kim by fastforce |
|
2288 have "T' \<subseteq> S" |
|
2289 by (meson opeT' subsetD openin_imp_subset) |
|
2290 then show "\<forall>x\<in>T'. k (0, x) = f x" "\<forall>x\<in>T'. k (1, x) = g x" |
|
2291 by (auto simp: kh' h'_def) |
|
2292 qed |
|
2293 ultimately show ?thesis |
|
2294 by (blast intro: that) |
|
2295 qed |
|
2296 |
|
2297 text\<open> Homotopy on a union of closed-open sets.\<close> |
|
2298 proposition\<^marker>\<open>tag unimportant\<close> homotopic_on_clopen_Union: |
|
2299 fixes \<F> :: "'a::euclidean_space set set" |
|
2300 assumes "\<And>S. S \<in> \<F> \<Longrightarrow> closedin (top_of_set (\<Union>\<F>)) S" |
|
2301 and "\<And>S. S \<in> \<F> \<Longrightarrow> openin (top_of_set (\<Union>\<F>)) S" |
|
2302 and "\<And>S. S \<in> \<F> \<Longrightarrow> homotopic_with_canon (\<lambda>x. True) S T f g" |
|
2303 shows "homotopic_with_canon (\<lambda>x. True) (\<Union>\<F>) T f g" |
|
2304 proof - |
|
2305 obtain \<V> where "\<V> \<subseteq> \<F>" "countable \<V>" and eqU: "\<Union>\<V> = \<Union>\<F>" |
|
2306 using Lindelof_openin assms by blast |
|
2307 show ?thesis |
|
2308 proof (cases "\<V> = {}") |
|
2309 case True |
|
2310 then show ?thesis |
|
2311 by (metis Union_empty eqU homotopic_with_canon_on_empty) |
|
2312 next |
|
2313 case False |
|
2314 then obtain V :: "nat \<Rightarrow> 'a set" where V: "range V = \<V>" |
|
2315 using range_from_nat_into \<open>countable \<V>\<close> by metis |
|
2316 with \<open>\<V> \<subseteq> \<F>\<close> have clo: "\<And>n. closedin (top_of_set (\<Union>\<F>)) (V n)" |
|
2317 and ope: "\<And>n. openin (top_of_set (\<Union>\<F>)) (V n)" |
|
2318 and hom: "\<And>n. homotopic_with_canon (\<lambda>x. True) (V n) T f g" |
|
2319 using assms by auto |
|
2320 then obtain h where conth: "\<And>n. continuous_on ({0..1::real} \<times> V n) (h n)" |
|
2321 and him: "\<And>n. h n ` ({0..1} \<times> V n) \<subseteq> T" |
|
2322 and h0: "\<And>n. \<And>x. x \<in> V n \<Longrightarrow> h n (0, x) = f x" |
|
2323 and h1: "\<And>n. \<And>x. x \<in> V n \<Longrightarrow> h n (1, x) = g x" |
|
2324 by (simp add: homotopic_with) metis |
|
2325 have wop: "b \<in> V x \<Longrightarrow> \<exists>k. b \<in> V k \<and> (\<forall>j<k. b \<notin> V j)" for b x |
|
2326 using nat_less_induct [where P = "\<lambda>i. b \<notin> V i"] by meson |
|
2327 obtain \<zeta> where cont: "continuous_on ({0..1} \<times> \<Union>(V ` UNIV)) \<zeta>" |
|
2328 and eq: "\<And>x i. \<lbrakk>x \<in> {0..1} \<times> \<Union>(V ` UNIV) \<inter> |
|
2329 {0..1} \<times> (V i - (\<Union>m<i. V m))\<rbrakk> \<Longrightarrow> \<zeta> x = h i x" |
|
2330 proof (rule pasting_lemma_exists) |
|
2331 let ?X = "top_of_set ({0..1::real} \<times> \<Union>(range V))" |
|
2332 show "topspace ?X \<subseteq> (\<Union>i. {0..1::real} \<times> (V i - (\<Union>m<i. V m)))" |
|
2333 by (force simp: Ball_def dest: wop) |
|
2334 show "openin (top_of_set ({0..1} \<times> \<Union>(V ` UNIV))) |
|
2335 ({0..1::real} \<times> (V i - (\<Union>m<i. V m)))" for i |
|
2336 proof (intro openin_Times openin_subtopology_self openin_diff) |
|
2337 show "openin (top_of_set (\<Union>(V ` UNIV))) (V i)" |
|
2338 using ope V eqU by auto |
|
2339 show "closedin (top_of_set (\<Union>(V ` UNIV))) (\<Union>m<i. V m)" |
|
2340 using V clo eqU by (force intro: closedin_Union) |
|
2341 qed |
|
2342 show "continuous_map (subtopology ?X ({0..1} \<times> (V i - \<Union> (V ` {..<i})))) euclidean (h i)" for i |
|
2343 by (auto simp add: subtopology_subtopology intro!: continuous_on_subset [OF conth]) |
|
2344 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)) |
|
2345 \<Longrightarrow> h i x = h j x" |
|
2346 by clarsimp (metis lessThan_iff linorder_neqE_nat) |
|
2347 qed auto |
|
2348 show ?thesis |
|
2349 proof (simp add: homotopic_with eqU [symmetric], intro exI conjI ballI) |
|
2350 show "continuous_on ({0..1} \<times> \<Union>\<V>) \<zeta>" |
|
2351 using V eqU by (blast intro!: continuous_on_subset [OF cont]) |
|
2352 show "\<zeta>` ({0..1} \<times> \<Union>\<V>) \<subseteq> T" |
|
2353 proof clarsimp |
|
2354 fix t :: real and y :: "'a" and X :: "'a set" |
|
2355 assume "y \<in> X" "X \<in> \<V>" and t: "0 \<le> t" "t \<le> 1" |
|
2356 then obtain k where "y \<in> V k" and j: "\<forall>j<k. y \<notin> V j" |
|
2357 by (metis image_iff V wop) |
|
2358 with him t show "\<zeta>(t, y) \<in> T" |
|
2359 by (subst eq) force+ |
|
2360 qed |
|
2361 fix X y |
|
2362 assume "X \<in> \<V>" "y \<in> X" |
|
2363 then obtain k where "y \<in> V k" and j: "\<forall>j<k. y \<notin> V j" |
|
2364 by (metis image_iff V wop) |
|
2365 then show "\<zeta>(0, y) = f y" and "\<zeta>(1, y) = g y" |
|
2366 by (subst eq [where i=k]; force simp: h0 h1)+ |
|
2367 qed |
|
2368 qed |
|
2369 qed |
|
2370 |
|
2371 lemma homotopic_on_components_eq: |
|
2372 fixes S :: "'a :: euclidean_space set" and T :: "'b :: euclidean_space set" |
|
2373 assumes S: "locally connected S \<or> compact S" and "ANR T" |
|
2374 shows "homotopic_with_canon (\<lambda>x. True) S T f g \<longleftrightarrow> |
|
2375 (continuous_on S f \<and> f ` S \<subseteq> T \<and> continuous_on S g \<and> g ` S \<subseteq> T) \<and> |
|
2376 (\<forall>C \<in> components S. homotopic_with_canon (\<lambda>x. True) C T f g)" |
|
2377 (is "?lhs \<longleftrightarrow> ?C \<and> ?rhs") |
|
2378 proof - |
|
2379 have "continuous_on S f" "f ` S \<subseteq> T" "continuous_on S g" "g ` S \<subseteq> T" if ?lhs |
|
2380 using homotopic_with_imp_continuous homotopic_with_imp_subset1 homotopic_with_imp_subset2 that by blast+ |
|
2381 moreover have "?lhs \<longleftrightarrow> ?rhs" |
|
2382 if contf: "continuous_on S f" and fim: "f ` S \<subseteq> T" and contg: "continuous_on S g" and gim: "g ` S \<subseteq> T" |
|
2383 proof |
|
2384 assume ?lhs |
|
2385 with that show ?rhs |
|
2386 by (simp add: homotopic_with_subset_left in_components_subset) |
|
2387 next |
|
2388 assume R: ?rhs |
|
2389 have "\<exists>U. C \<subseteq> U \<and> closedin (top_of_set S) U \<and> |
|
2390 openin (top_of_set S) U \<and> |
|
2391 homotopic_with_canon (\<lambda>x. True) U T f g" if C: "C \<in> components S" for C |
|
2392 proof - |
|
2393 have "C \<subseteq> S" |
|
2394 by (simp add: in_components_subset that) |
|
2395 show ?thesis |
|
2396 using S |
|
2397 proof |
|
2398 assume "locally connected S" |
|
2399 show ?thesis |
|
2400 proof (intro exI conjI) |
|
2401 show "closedin (top_of_set S) C" |
|
2402 by (simp add: closedin_component that) |
|
2403 show "openin (top_of_set S) C" |
|
2404 by (simp add: \<open>locally connected S\<close> openin_components_locally_connected that) |
|
2405 show "homotopic_with_canon (\<lambda>x. True) C T f g" |
|
2406 by (simp add: R that) |
|
2407 qed auto |
|
2408 next |
|
2409 assume "compact S" |
|
2410 have hom: "homotopic_with_canon (\<lambda>x. True) C T f g" |
|
2411 using R that by blast |
|
2412 obtain U where "C \<subseteq> U" and opeU: "openin (top_of_set S) U" |
|
2413 and hom: "homotopic_with_canon (\<lambda>x. True) U T f g" |
|
2414 using homotopic_neighbourhood_extension [OF contf fim contg gim _ \<open>ANR T\<close> hom] |
|
2415 \<open>C \<in> components S\<close> closedin_component by blast |
|
2416 then obtain V where "open V" and V: "U = S \<inter> V" |
|
2417 by (auto simp: openin_open) |
|
2418 moreover have "locally compact S" |
|
2419 by (simp add: \<open>compact S\<close> closed_imp_locally_compact compact_imp_closed) |
|
2420 ultimately obtain K where opeK: "openin (top_of_set S) K" and "compact K" "C \<subseteq> K" "K \<subseteq> V" |
|
2421 by (metis C Int_subset_iff Sura_Bura_clopen_subset \<open>C \<subseteq> U\<close> \<open>compact S\<close> compact_components) |
|
2422 show ?thesis |
|
2423 proof (intro exI conjI) |
|
2424 show "closedin (top_of_set S) K" |
|
2425 by (meson \<open>compact K\<close> \<open>compact S\<close> closedin_compact_eq opeK openin_imp_subset) |
|
2426 show "homotopic_with_canon (\<lambda>x. True) K T f g" |
|
2427 using V \<open>K \<subseteq> V\<close> hom homotopic_with_subset_left opeK openin_imp_subset by fastforce |
|
2428 qed (use opeK \<open>C \<subseteq> K\<close> in auto) |
|
2429 qed |
|
2430 qed |
|
2431 then obtain \<phi> where \<phi>: "\<And>C. C \<in> components S \<Longrightarrow> C \<subseteq> \<phi> C" |
|
2432 and clo\<phi>: "\<And>C. C \<in> components S \<Longrightarrow> closedin (top_of_set S) (\<phi> C)" |
|
2433 and ope\<phi>: "\<And>C. C \<in> components S \<Longrightarrow> openin (top_of_set S) (\<phi> C)" |
|
2434 and hom\<phi>: "\<And>C. C \<in> components S \<Longrightarrow> homotopic_with_canon (\<lambda>x. True) (\<phi> C) T f g" |
|
2435 by metis |
|
2436 have Seq: "S = \<Union> (\<phi> ` components S)" |
|
2437 proof |
|
2438 show "S \<subseteq> \<Union> (\<phi> ` components S)" |
|
2439 by (metis Sup_mono Union_components \<phi> imageI) |
|
2440 show "\<Union> (\<phi> ` components S) \<subseteq> S" |
|
2441 using ope\<phi> openin_imp_subset by fastforce |
|
2442 qed |
|
2443 show ?lhs |
|
2444 apply (subst Seq) |
|
2445 apply (rule homotopic_on_clopen_Union) |
|
2446 using Seq clo\<phi> ope\<phi> hom\<phi> by auto |
|
2447 qed |
|
2448 ultimately show ?thesis by blast |
|
2449 qed |
|
2450 |
|
2451 |
|
2452 lemma cohomotopically_trivial_on_components: |
|
2453 fixes S :: "'a :: euclidean_space set" and T :: "'b :: euclidean_space set" |
|
2454 assumes S: "locally connected S \<or> compact S" and "ANR T" |
|
2455 shows |
|
2456 "(\<forall>f g. continuous_on S f \<longrightarrow> f ` S \<subseteq> T \<longrightarrow> continuous_on S g \<longrightarrow> g ` S \<subseteq> T \<longrightarrow> |
|
2457 homotopic_with_canon (\<lambda>x. True) S T f g) |
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2458 \<longleftrightarrow> |
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2459 (\<forall>C\<in>components S. |
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2460 \<forall>f g. continuous_on C f \<longrightarrow> f ` C \<subseteq> T \<longrightarrow> continuous_on C g \<longrightarrow> g ` C \<subseteq> T \<longrightarrow> |
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2461 homotopic_with_canon (\<lambda>x. True) C T f g)" |
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2462 (is "?lhs = ?rhs") |
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2463 proof |
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2464 assume L[rule_format]: ?lhs |
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2465 show ?rhs |
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2466 proof clarify |
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2467 fix C f g |
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2468 assume contf: "continuous_on C f" and fim: "f ` C \<subseteq> T" |
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2469 and contg: "continuous_on C g" and gim: "g ` C \<subseteq> T" and C: "C \<in> components S" |
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2470 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" |
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2471 using extension_from_component [OF S \<open>ANR T\<close> C contf fim] by metis |
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2472 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" |
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2473 using extension_from_component [OF S \<open>ANR T\<close> C contg gim] by metis |
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2474 have "homotopic_with_canon (\<lambda>x. True) C T f' g'" |
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2475 using L [OF contf' f'im contg' g'im] homotopic_with_subset_left C in_components_subset by fastforce |
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2476 then show "homotopic_with_canon (\<lambda>x. True) C T f g" |
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2477 using f'f g'g homotopic_with_eq by force |
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2478 qed |
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2479 next |
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2480 assume R [rule_format]: ?rhs |
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2481 show ?lhs |
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2482 proof clarify |
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2483 fix f g |
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2484 assume contf: "continuous_on S f" and fim: "f ` S \<subseteq> T" |
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2485 and contg: "continuous_on S g" and gim: "g ` S \<subseteq> T" |
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2486 moreover have "homotopic_with_canon (\<lambda>x. True) C T f g" if "C \<in> components S" for C |
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2487 using R [OF that] |
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2488 by (meson contf contg continuous_on_subset fim gim image_mono in_components_subset order.trans that) |
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2489 ultimately show "homotopic_with_canon (\<lambda>x. True) S T f g" |
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2490 by (subst homotopic_on_components_eq [OF S \<open>ANR T\<close>]) auto |
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2491 qed |
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2492 qed |
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2493 |
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2494 subsection\<open>The complement of a set and path-connectedness\<close> |
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2495 |
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2496 text\<open>Complement in dimension N > 1 of set homeomorphic to any interval in |
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2497 any dimension is (path-)connected. This naively generalizes the argument |
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2498 in Ryuji Maehara's paper "The Jordan curve theorem via the Brouwer fixed point theorem", |
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2499 American Mathematical Monthly 1984.\<close> |
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2500 |
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2501 lemma unbounded_components_complement_absolute_retract: |
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2502 fixes S :: "'a::euclidean_space set" |
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2503 assumes C: "C \<in> components(- S)" and S: "compact S" "AR S" |
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2504 shows "\<not> bounded C" |
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2505 proof - |
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2506 obtain y where y: "C = connected_component_set (- S) y" and "y \<notin> S" |
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2507 using C by (auto simp: components_def) |
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2508 have "open(- S)" |
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2509 using S by (simp add: closed_open compact_eq_bounded_closed) |
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2510 have "S retract_of UNIV" |
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2511 using S compact_AR by blast |
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2512 then obtain r where contr: "continuous_on UNIV r" and ontor: "range r \<subseteq> S" |
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2513 and r: "\<And>x. x \<in> S \<Longrightarrow> r x = x" |
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2514 by (auto simp: retract_of_def retraction_def) |
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2515 show ?thesis |
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2516 proof |
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2517 assume "bounded C" |
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2518 have "connected_component_set (- S) y \<subseteq> S" |
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2519 proof (rule frontier_subset_retraction) |
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2520 show "bounded (connected_component_set (- S) y)" |
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2521 using \<open>bounded C\<close> y by blast |
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2522 show "frontier (connected_component_set (- S) y) \<subseteq> S" |
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2523 using C \<open>compact S\<close> compact_eq_bounded_closed frontier_of_components_closed_complement y by blast |
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2524 show "continuous_on (closure (connected_component_set (- S) y)) r" |
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2525 by (blast intro: continuous_on_subset [OF contr]) |
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2526 qed (use ontor r in auto) |
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2527 with \<open>y \<notin> S\<close> show False by force |
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2528 qed |
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2529 qed |
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2530 |
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2531 lemma connected_complement_absolute_retract: |
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2532 fixes S :: "'a::euclidean_space set" |
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2533 assumes S: "compact S" "AR S" and 2: "2 \<le> DIM('a)" |
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2534 shows "connected(- S)" |
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2535 proof - |
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2536 have "S retract_of UNIV" |
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2537 using S compact_AR by blast |
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2538 show ?thesis |
|
2539 apply (clarsimp simp: connected_iff_connected_component_eq) |
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2540 apply (rule cobounded_unique_unbounded_component [OF _ 2]) |
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2541 apply (simp add: \<open>compact S\<close> compact_imp_bounded) |
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2542 apply (meson ComplI S componentsI unbounded_components_complement_absolute_retract)+ |
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2543 done |
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2544 qed |
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2545 |
|
2546 lemma path_connected_complement_absolute_retract: |
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2547 fixes S :: "'a::euclidean_space set" |
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2548 assumes "compact S" "AR S" "2 \<le> DIM('a)" |
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2549 shows "path_connected(- S)" |
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2550 using connected_complement_absolute_retract [OF assms] |
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2551 using \<open>compact S\<close> compact_eq_bounded_closed connected_open_path_connected by blast |
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2552 |
|
2553 theorem connected_complement_homeomorphic_convex_compact: |
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2554 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
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2555 assumes hom: "S homeomorphic T" and T: "convex T" "compact T" and 2: "2 \<le> DIM('a)" |
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2556 shows "connected(- S)" |
|
2557 proof (cases "S = {}") |
|
2558 case True |
|
2559 then show ?thesis |
|
2560 by (simp add: connected_UNIV) |
|
2561 next |
|
2562 case False |
|
2563 show ?thesis |
|
2564 proof (rule connected_complement_absolute_retract) |
|
2565 show "compact S" |
|
2566 using \<open>compact T\<close> hom homeomorphic_compactness by auto |
|
2567 show "AR S" |
|
2568 by (meson AR_ANR False \<open>convex T\<close> convex_imp_ANR convex_imp_contractible hom homeomorphic_ANR_iff_ANR homeomorphic_contractible_eq) |
|
2569 qed (rule 2) |
|
2570 qed |
|
2571 |
|
2572 corollary path_connected_complement_homeomorphic_convex_compact: |
|
2573 fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
|
2574 assumes hom: "S homeomorphic T" "convex T" "compact T" "2 \<le> DIM('a)" |
|
2575 shows "path_connected(- S)" |
|
2576 using connected_complement_homeomorphic_convex_compact [OF assms] |
|
2577 using \<open>compact T\<close> compact_eq_bounded_closed connected_open_path_connected hom homeomorphic_compactness by blast |
|
2578 |
|
2579 lemma path_connected_complement_homeomorphic_interval: |
|
2580 fixes S :: "'a::euclidean_space set" |
|
2581 assumes "S homeomorphic cbox a b" "2 \<le> DIM('a)" |
|
2582 shows "path_connected(-S)" |
|
2583 using assms compact_cbox convex_box(1) path_connected_complement_homeomorphic_convex_compact by blast |
|
2584 |
|
2585 lemma connected_complement_homeomorphic_interval: |
|
2586 fixes S :: "'a::euclidean_space set" |
|
2587 assumes "S homeomorphic cbox a b" "2 \<le> DIM('a)" |
|
2588 shows "connected(-S)" |
|
2589 using assms path_connected_complement_homeomorphic_interval path_connected_imp_connected by blast |
|
2590 |
|
2591 end |