Quelle Urysohn.thy Sprache: Isabelle

(* Title: HOL/Analysis/Urysohn.thy
 Authors: LC Paulson, based on material from HOL Light
*)

section \<open>The Urysohn lemma, its consequences and other advanced material about metric spaces\<close>

theory Urysohn
imports Abstract_Topological_Spaces Abstract_Metric_Spaces Infinite_Sum Arcwise_Connected
begin

subsection \<open>Urysohn lemma and Tietze's theorem\<close>

proposition Urysohn_lemma:
 fixes a b :: real
 assumes "normal_space X" "closedin X S" "closedin X T" "disjnt S T" "a \ b"
 obtains f where "continuous_map X (top_of_set {a..b}) f" "f ` S \ {a}" "f ` T \ {b}"
proof -
 obtain U where "openin X U" "S \ U" "X closure_of U \ topspace X - T"
 using assms unfolding normal_space_alt disjnt_def
 by (metis Diff_mono Un_Diff_Int closedin_def subset_eq sup_bot_right)
 have "\G :: real \ 'a set. G 0 = U \ G 1 = topspace X - T \
 (\<forall>x \<in> dyadics \<inter> {0..1}. \<forall>y \<in> dyadics \<inter> {0..1}. x < y \<longrightarrow> openin X (G x) \<and> openin X (G y) \<and> X closure_of (G x) \<subseteq> G y)"
 proof (rule recursion_on_dyadic_fractions)
 show "openin X U \ openin X (topspace X - T) \ X closure_of U \ topspace X - T"
 using \<open>X closure_of U \<subseteq> topspace X - T\<close> \<open>openin X U\<close> \<open>closedin X T\<close> by blast
 show "\z. (openin X x \ openin X z \ X closure_of x \ z) \ openin X z \ openin X y \ X closure_of z \ y"
 if "openin X x \ openin X y \ X closure_of x \ y" for x y
 by (meson that closedin_closure_of normal_space_alt \<open>normal_space X\<close>)
 show "openin X x \ openin X z \ X closure_of x \ z"
 if "openin X x \ openin X y \ X closure_of x \ y" and "openin X y \ openin X z \ X closure_of y \ z" for x y z
 by (meson that closure_of_subset openin_subset subset_trans)
 qed
 then obtain G :: "real \ 'a set"
 where G0: "G 0 = U" and G1: "G 1 = topspace X - T"
 and G: "\x y. \x \ dyadics; y \ dyadics; 0 \ x; x < y; y \ 1\
 \<Longrightarrow> openin X (G x) \<and> openin X (G y) \<and> X closure_of (G x) \<subseteq> G y"
 by (smt (verit, del_insts) Int_iff atLeastAtMost_iff)
 define f where "f \ \x. Inf(insert 1 {r. r \ dyadics \ {0..1} \ x \ G r})"
 have f_ge: "f x \ 0" if "x \ topspace X" for x
 unfolding f_def by (force intro: cInf_greatest)
 moreover have f_le1: "f x \ 1" if "x \ topspace X" for x
 proof -
 have "bdd_below {r \ dyadics \ {0..1}. x \ G r}"
 by (auto simp: bdd_below_def)
 then show ?thesis
 by (auto simp: f_def cInf_lower)
 qed
 ultimately have fim: "f \ topspace X \ {0..1}"
 by (auto simp: f_def)
 have 0: "0 \ dyadics \ {0..1::real}" and 1: "1 \ dyadics \ {0..1::real}"
 by (force simp: dyadics_def)+
 then have opeG: "openin X (G r)" if "r \ dyadics \ {0..1}" for r
 using G G0 \<open>openin X U\<close> less_eq_real_def that by auto
 have "x \ G 0" if "x \ S" for x
 using G0 \<open>S \<subseteq> U\<close> that by blast
 with 0 have fimS: "f ` S \ {0}"
 unfolding f_def by (force intro!: cInf_eq_minimum)
 have False if "r \ dyadics" "0 \ r" "r < 1" "x \ G r" "x \ T" for r x
 using G [of r 1] 1
 by (smt (verit, best) DiffD2 G1 Int_iff closure_of_subset inf.orderE openin_subset that)
 then have "r\1" if "r \ dyadics" "0 \ r" "r \ 1" "x \ G r" "x \ T" for r x
 using linorder_not_le that by blast
 then have fimT: "f ` T \ {1}"
 unfolding f_def by (force intro!: cInf_eq_minimum)
 have fle1: "f z \ 1" for z
 by (force simp: f_def intro: cInf_lower)
 have fle: "f z \ x" if "x \ dyadics \ {0..1}" "z \ G x" for z x
 using that by (force simp: f_def intro: cInf_lower)
 have *: "b \ f z" if "b \ 1" "\x. \x \ dyadics \ {0..1}; z \ G x\ \ b \ x" for z b
 using that by (force simp: f_def intro: cInf_greatest)
 have **: "r \ f x" if r: "r \ dyadics \ {0..1}" "x \ G r" for r x
 proof (rule *)
 show "r \ s" if "s \ dyadics \ {0..1}" and "x \ G s" for s :: real
 using that r G [of s r] by (force simp: dest: closure_of_subset openin_subset)
 qed (use that in force)

 have "\U. openin X U \ x \ U \ (\y \ U. \f y - f x\ < \)"
 if "x \ topspace X" and "0 < \" for x \
 proof -
 have A: "\r. r \ dyadics \ {0..1} \ r < y \ \r - y\ < d" if "0 < y" "y \ 1" "0 < d" for y d::real
 proof -
 obtain n q r
 where "of_nat q / 2^n < y" "y < of_nat r / 2^n" "\q / 2^n - r / 2^n \ < d"
 by (smt (verit, del_insts) padic_rational_approximation_straddle_pos \<open>0 < d\<close> \<open>0 < y\<close>)
 then show ?thesis
 unfolding dyadics_def
 using divide_eq_0_iff that(2) by fastforce
 qed
 have B: "\r. r \ dyadics \ {0..1} \ y < r \ \r - y\ < d" if "0 \ y" "y < 1" "0 < d" for y d::real
 proof -
 obtain n q r
 where "of_nat q / 2^n \ y" "y < of_nat r / 2^n" "\q / 2^n - r / 2^n \ < d"
 using padic_rational_approximation_straddle_pos_le
 by (smt (verit, del_insts) \<open>0 < d\<close> \<open>0 \<le> y\<close>)
 then show ?thesis
 apply (clarsimp simp: dyadics_def)
 using divide_eq_0_iff \<open>y < 1\<close>
 by (smt (verit) divide_nonneg_nonneg divide_self of_nat_0_le_iff of_nat_1 power_0 zero_le_power)
 qed
 show ?thesis
 proof (cases "f x = 0")
 case True
 with B[of 0] obtain r where r: "r \ dyadics \ {0..1}" "0 < r" "\r\ < \/2"
 by (smt (verit) \<open>0 < \<epsilon>\<close> half_gt_zero)
 show ?thesis
 proof (intro exI conjI)
 show "openin X (G r)"
 using opeG r(1) by blast
 show "x \ G r"
 using True ** r by force
 show "\y \ G r. \f y - f x\ < \"
 using f_ge \<open>openin X (G r)\<close> fle openin_subset r by (fastforce simp: True)
 qed
 next
 case False
 show ?thesis
 proof (cases "f x = 1")
 case True
 with A[of 1] obtain r where r: "r \ dyadics \ {0..1}" "r < 1" "\r-1\ < \/2"
 by (smt (verit) \<open>0 < \<epsilon>\<close> half_gt_zero)
 define G' where "G' \<equiv> topspace X - X closure_of G r"
 show ?thesis
 proof (intro exI conjI)
 show "openin X G'"
 unfolding G'_def by fastforce
 obtain r' where "r' \<in> dyadics \<and> 0 \<le> r' \<and> r' \<le> 1 \<and> r < r' \<and> \<bar>r' - r\<bar> < 1 - r"
 using B r by force
 moreover
 have "1 - r \ dyadics" "0 \ r"
 using 1 r dyadics_diff by force+
 ultimately have "x \ X closure_of G r"
 using G True r fle by force
 then show "x \ G'"
 by (simp add: G'_def that)
 show "\y \ G'. \f y - f x\ < \"
 using ** f_le1 in_closure_of r by (fastforce simp: True G'_def)
 qed
 next
 case False
 have "0 < f x" "f x < 1"
 using fle1 f_ge that(1) \<open>f x \<noteq> 0\<close> \<open>f x \<noteq> 1\<close> by (metis order_le_less) +
 obtain r where r: "r \ dyadics \ {0..1}" "r < f x" "\r - f x\ < \ / 2"
 using A \<open>0 < \<epsilon>\<close> \<open>0 < f x\<close> \<open>f x < 1\<close> by (smt (verit, best) half_gt_zero)
 obtain r' where r': "r' \ dyadics \ {0..1}" "f x < r'" "\r' - f x\ < \ / 2"
 using B \<open>0 < \<epsilon>\<close> \<open>0 < f x\<close> \<open>f x < 1\<close> by (smt (verit, best) half_gt_zero)
 have "r < 1"
 using \<open>f x < 1\<close> r(2) by force
 show ?thesis
 proof (intro conjI exI)
 show "openin X (G r' - X closure_of G r)"
 using closedin_closure_of opeG r' by blast
 have "x \ X closure_of G r \ False"
 using B [of r "f x - r"] r \<open>r < 1\<close> G [of r] fle by force
 then show "x \ G r' - X closure_of G r"
 using ** r' by fastforce
 show "\y\G r' - X closure_of G r. \f y - f x\ < \"
 using r r' ** G closure_of_subset field_sum_of_halves fle openin_subset subset_eq
 by (smt (verit) DiffE opeG)
 qed
 qed
 qed
 qed
 then have contf: "continuous_map X (top_of_set {0..1}) f"
 by (auto simp: Met_TC.continuous_map_to_metric dist_real_def continuous_map_in_subtopology fim abs_minus_commute simp flip: mtopology_is_euclidean)
 define g where "g \ \x. a + (b - a) * f x"
 show thesis
 proof
 have "continuous_map X euclideanreal g"
 using contf \<open>a \<le> b\<close> unfolding g_def by (auto simp: continuous_intros continuous_map_in_subtopology)
 moreover have "g \ (topspace X) \ {a..b}"
 using mult_left_le [of "f _" "b-a"] contf \<open>a \<le> b\<close>
 by (simp add: g_def Pi_iff add.commute continuous_map_in_subtopology image_subset_iff le_diff_eq)
 ultimately show "continuous_map X (top_of_set {a..b}) g"
 using continuous_map_in_subtopology by blast
 show "g ` S \ {a}" "g ` T \ {b}"
 using fimS fimT by (auto simp: g_def)
 qed
qed

lemma Urysohn_lemma_alt:
 fixes a b :: real
 assumes "normal_space X" "closedin X S" "closedin X T" "disjnt S T"
 obtains f where "continuous_map X euclideanreal f" "f ` S \ {a}" "f ` T \ {b}"
 by (metis Urysohn_lemma assms continuous_map_in_subtopology disjnt_sym linear)

lemma normal_space_iff_Urysohn_gen_alt:
 assumes "a \ b"
 shows "normal_space X \
 (\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T
 \<longrightarrow> (\<exists>f. continuous_map X euclideanreal f \<and> f ` S \<subseteq> {a} \<and> f ` T \<subseteq> {b}))"
(is "?lhs=?rhs")
proof
 show "?lhs \ ?rhs"
 by (metis Urysohn_lemma_alt)
next
 assume R: ?rhs
 show ?lhs
 unfolding normal_space_def
 proof clarify
 fix S T
 assume "closedin X S" and "closedin X T" and "disjnt S T"
 with R obtain f where contf: "continuous_map X euclideanreal f" and "f ` S \ {a}" "f ` T \ {b}"
 by meson
 show "\U V. openin X U \ openin X V \ S \ U \ T \ V \ disjnt U V"
 proof (intro conjI exI)
 show "openin X {x \ topspace X. f x \ ball a (\a - b\ / 2)}"
 by (force intro!: openin_continuous_map_preimage [OF contf])
 show "openin X {x \ topspace X. f x \ ball b (\a - b\ / 2)}"
 by (force intro!: openin_continuous_map_preimage [OF contf])
 show "S \ {x \ topspace X. f x \ ball a (\a - b\ / 2)}"
 using \<open>closedin X S\<close> closedin_subset \<open>f ` S \<subseteq> {a}\<close> assms by force
 show "T \ {x \ topspace X. f x \ ball b (\a - b\ / 2)}"
 using \<open>closedin X T\<close> closedin_subset \<open>f ` T \<subseteq> {b}\<close> assms by force
 have "\x. \x \ topspace X; dist a (f x) < \a-b\/2; dist b (f x) < \a-b\/2\ \ False"
 by (smt (verit, best) dist_real_def dist_triangle_half_l)
 then show "disjnt {x \ topspace X. f x \ ball a (\a-b\ / 2)} {x \ topspace X. f x \ ball b (\a-b\ / 2)}"
 using disjnt_iff by fastforce
 qed
 qed
qed

lemma normal_space_iff_Urysohn_gen:
 fixes a b::real
 shows
 "a 
 (\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T
 \<longrightarrow> (\<exists>f. continuous_map X (top_of_set {a..b}) f \<and>
 f ` S \<subseteq> {a} \<and> f ` T \<subseteq> {b}))"
 by (metis linear not_le Urysohn_lemma normal_space_iff_Urysohn_gen_alt continuous_map_in_subtopology)

lemma normal_space_iff_Urysohn_alt:
 "normal_space X \
 (\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T
 \<longrightarrow> (\<exists>f. continuous_map X euclideanreal f \<and>
 f ` S \<subseteq> {0} \<and> f ` T \<subseteq> {1}))"
 by (rule normal_space_iff_Urysohn_gen_alt) auto

lemma normal_space_iff_Urysohn:
 "normal_space X \
 (\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T
 \<longrightarrow> (\<exists>f::'a\<Rightarrow>real. continuous_map X (top_of_set {0..1}) f \<and>
 f ` S \<subseteq> {0} \<and> f ` T \<subseteq> {1}))"
 by (rule normal_space_iff_Urysohn_gen) auto

lemma normal_space_perfect_map_image:
 "\normal_space X; perfect_map X Y f\ \ normal_space Y"
 unfolding perfect_map_def proper_map_def
 using normal_space_continuous_closed_map_image by fastforce

lemma Hausdorff_normal_space_closed_continuous_map_image:
 "\normal_space X; closed_map X Y f; continuous_map X Y f;
 f ` topspace X = topspace Y; t1_space Y\<rbrakk>
 \<Longrightarrow> Hausdorff_space Y"
 by (metis normal_space_continuous_closed_map_image normal_t1_imp_Hausdorff_space)

lemma normal_Hausdorff_space_closed_continuous_map_image:
 "\normal_space X; Hausdorff_space X; closed_map X Y f;
 continuous_map X Y f; f ` topspace X = topspace Y\<rbrakk>
 \<Longrightarrow> normal_space Y \<and> Hausdorff_space Y"
 by (meson normal_space_continuous_closed_map_image normal_t1_eq_Hausdorff_space t1_space_closed_map_image)

lemma Lindelof_cover:
 assumes "regular_space X" and "Lindelof_space X" and "S \ {}"
 and clo: "closedin X S" "closedin X T" "disjnt S T"
 obtains h :: "nat \ 'a set" where
 "\n. openin X (h n)" "\n. disjnt T (X closure_of (h n))" and "S \ \(range h)"
proof -
 have "\U. openin X U \ x \ U \ disjnt T (X closure_of U)"
 if "x \ S" for x
 using \<open>regular_space X\<close> unfolding regular_space
 by (metis (full_types) Diff_iff \<open>disjnt S T\<close> clo closure_of_eq disjnt_iff in_closure_of that)
 then obtain h where oh: "\x. x \ S \ openin X (h x)"
 and xh: "\x. x \ S \ x \ h x"
 and dh: "\x. x \ S \ disjnt T (X closure_of h x)"
 by metis
 have "Lindelof_space(subtopology X S)"
 by (simp add: Lindelof_space_closedin_subtopology \<open>Lindelof_space X\<close> \<open>closedin X S\<close>)
 then obtain \ where \: "countable \ \<and> \ \<subseteq> h ` S \<and> S \<subseteq> \<Union>\"
 unfolding Lindelof_space_subtopology_subset [OF closedin_subset [OF \<open>closedin X S\<close>]]
 by (smt (verit, del_insts) oh xh UN_I image_iff subsetI)
 with \<open>S \<noteq> {}\<close> have "\ \<noteq> {}"
 by blast
 show ?thesis
 proof
 show "openin X (from_nat_into \ n)" for n
 by (metis \ from_nat_into image_iff \<open>\ \<noteq> {}\<close> oh subsetD)
 show "disjnt T (X closure_of (from_nat_into \) n)" for n
 using dh from_nat_into [OF \<open>\ \<noteq> {}\<close>]
 by (metis \ f_inv_into_f inv_into_into subset_eq)
 show "S \ \(range (from_nat_into \))"
 by (simp add: \ \<open>\ \<noteq> {}\<close>)
 qed
qed

lemma regular_Lindelof_imp_normal_space:
 assumes "regular_space X" and "Lindelof_space X"
 shows "normal_space X"
 unfolding normal_space_def
proof clarify
 fix S T
 assume clo: "closedin X S" "closedin X T" and "disjnt S T"
 show "\U V. openin X U \ openin X V \ S \ U \ T \ V \ disjnt U V"
 proof (cases "S={} \ T={}")
 case True
 with clo show ?thesis
 by (meson closedin_def disjnt_empty1 disjnt_empty2 openin_empty openin_topspace subset_empty)
 next
 case False
 obtain h :: "nat \ 'a set" where
 opeh: "\n. openin X (h n)" and dish: "\n. disjnt T (X closure_of (h n))"
 and Sh: "S \ \(range h)"
 by (metis Lindelof_cover False \<open>disjnt S T\<close> assms clo)
 obtain k :: "nat \ 'a set" where
 opek: "\n. openin X (k n)" and disk: "\n. disjnt S (X closure_of (k n))"
 and Tk: "T \ \(range k)"
 by (metis Lindelof_cover False \<open>disjnt S T\<close> assms clo disjnt_sym)
 define U where "U \ \i. h i - (\j
 define V where "V \ \i. k i - (\j\i. X closure_of h j)"
 show ?thesis
 proof (intro exI conjI)
 show "openin X U" "openin X V"
 unfolding U_def V_def
 by (force intro!: opek opeh closedin_Union closedin_closure_of)+
 show "S \ U" "T \ V"
 using Sh Tk dish disk by (fastforce simp: U_def V_def disjnt_iff)+
 have "\x i j. \x \ k i; x \ h j; \j\i. x \ X closure_of h j\
 \<Longrightarrow> \<exists>i<j. x \<in> X closure_of k i"
 by (metis in_closure_of linorder_not_less opek openin_subset subsetD)
 then show "disjnt U V"
 by (force simp: U_def V_def disjnt_iff)
 qed
 qed
qed

theorem Tietze_extension_closed_real_interval:
 assumes "normal_space X" and "closedin X S"
 and contf: "continuous_map (subtopology X S) euclideanreal f"
 and fim: "f ` S \ {a..b}" and "a \ b"
 obtains g
 where "continuous_map X euclideanreal g"
 "\x. x \ S \ g x = f x" "g ` topspace X \ {a..b}"
proof -
 define c where "c \ max \a\ \b\ + 1"
 have "0 < c" and c: "\x. x \ S \ \f x\ \ c"
 using fim by (auto simp: c_def image_subset_iff)
 define good where
 "good \ \g n. continuous_map X euclideanreal g \ (\x \ S. \f x - g x\ \ c * (2/3)^n)"
 have step: "\g. good g (Suc n) \
 (\<forall>x \<in> topspace X. \<bar>g x - h x\<bar> \<le> c * (2/3)^n / 3)"
 if h: "good h n" for n h
 proof -
 have pos: "0 < c * (2/3) ^ n"
 by (simp add: \<open>0 < c\<close>)
 have S_eq: "S = topspace(subtopology X S)" and "S \ topspace X"
 using \<open>closedin X S\<close> closedin_subset by auto
 define d where "d \ c/3 * (2/3) ^ n"
 define SA where "SA \ {x \ S. f x - h x \ {..-d}}"
 define SB where "SB \ {x \ S. f x - h x \ {d..}}"
 have contfh: "continuous_map (subtopology X S) euclideanreal (\x. f x - h x)"
 using that
 by (simp add: contf good_def continuous_map_diff continuous_map_from_subtopology)
 then have "closedin (subtopology X S) SA"
 unfolding SA_def continuous_map_closedin
 by (metis (full_types) S_eq closed_atMost closed_closedin)
 then have "closedin X SA"
 using \<open>closedin X S\<close> closedin_trans_full by blast
 moreover have "closedin (subtopology X S) SB"
 unfolding SB_def
 using closedin_continuous_map_preimage_gen [OF contfh]
 by (metis (full_types) S_eq closed_atLeast closed_closedin closedin_topspace)
 then have "closedin X SB"
 using \<open>closedin X S\<close> closedin_trans_full by blast
 moreover have "disjnt SA SB"
 using pos by (auto simp: d_def disjnt_def SA_def SB_def)
 moreover have "-d \ d"
 using pos by (auto simp: d_def)
 ultimately
 obtain g where contg: "continuous_map X (top_of_set {- d..d}) g"
 and ga: "g ` SA \ {- d}" and gb: "g ` SB \ {d}"
 using Urysohn_lemma \<open>normal_space X\<close> by metis
 then have g_le_d: "\x. x \ topspace X \ \g x\ \ d"
 by (fastforce simp: abs_le_iff continuous_map_def minus_le_iff)
 have g_eq_d: "\x. \x \ S; f x - h x \ -d\ \ g x = -d"
 using ga by (auto simp: SA_def)
 have g_eq_negd: "\x. \x \ S; f x - h x \ d\ \ g x = d"
 using gb by (auto simp: SB_def)
 show ?thesis
 unfolding good_def
 proof (intro conjI strip exI)
 show "continuous_map X euclideanreal (\x. h x + g x)"
 using contg continuous_map_add continuous_map_in_subtopology that
 unfolding good_def by blast
 show "\f x - (h x + g x)\ \ c * (2 / 3) ^ Suc n" if "x \ S" for x
 proof -
 have x: "x \ topspace X"
 using \<open>S \<subseteq> topspace X\<close> that by auto
 have "\f x - h x\ \ c * (2/3) ^ n"
 using good_def h that by blast
 with g_eq_d [OF that] g_eq_negd [OF that] g_le_d [OF x]
 have "\f x - (h x + g x)\ \ d + d"
 unfolding d_def by linarith
 then show ?thesis
 by (simp add: d_def)
 qed
 show "\h x + g x - h x\ \ c * (2 / 3) ^ n / 3" if "x \ topspace X" for x
 using that d_def g_le_d by auto
 qed
 qed
 then obtain nxtg where nxtg: "\h n. good h n \
 good (nxtg h n) (Suc n) \<and> (\<forall>x \<in> topspace X. \<bar>nxtg h n x - h x\<bar> \<le> c * (2/3)^n / 3)"
 by metis
 define g where "g \ rec_nat (\x. 0) (\n r. nxtg r n)"
 have [simp]: "g 0 x = 0" for x
 by (auto simp: g_def)
 have g_Suc: "g(Suc n) = nxtg (g n) n" for n
 by (auto simp: g_def)
 have good: "good (g n) n" for n
 proof (induction n)
 case 0
 with c show ?case
 by (auto simp: good_def)
 qed (simp add: g_Suc nxtg)
 have *: "\n x. x \ topspace X \ \g(Suc n) x - g n x\ \ c * (2/3) ^ n / 3"
 using nxtg g_Suc good by force
 obtain h where conth: "continuous_map X euclideanreal h"
 and h: "\\. 0 < \ \ \\<^sub>F n in sequentially. \x\topspace X. dist (g n x) (h x) < \"
 proof (rule Met_TC.continuous_map_uniformly_Cauchy_limit)
 show "\\<^sub>F n in sequentially. continuous_map X (Met_TC.mtopology) (g n)"
 using good good_def by fastforce
 show "\N. \m n x. N \ m \ N \ n \ x \ topspace X \ dist (g m x) (g n x) < \"
 if "\ > 0" for \
 proof -
 have "\\<^sub>F n in sequentially. \(2/3) ^ n\ < \/c"
 proof (rule Archimedean_eventually_pow_inverse)
 show "0 < \ / c"
 by (simp add: \<open>0 < c\<close> that)
 qed auto
 then obtain N where N: "\n. n \ N \ \(2/3) ^ n\ < \/c"
 by (meson eventually_sequentially order_le_less_trans)
 have "\g m x - g n x\ < \"
 if "N \ m" "N \ n" and x: "x \ topspace X" "m \ n" for m n x
 proof (cases "m < n")
 case True
 have 23: "(\k = m..
 using \<open>m \<le> n\<close>
 by (induction n) (auto simp: le_Suc_eq)
 have "\g m x - g n x\ \ \\k = m.."
 by (subst sum_Suc_diff' [OF \m \ n\]) linarith
 also have "\ \ (\k = m..g (Suc k) x - g k x\)"
 by (rule sum_abs)
 also have "\ \ (\k = m..
 by (meson "*" sum_mono x(1))
 also have "\ = (c/3) * (\k = m..
 by (simp add: sum_distrib_left)
 also have "\ = (c/3) * 3 * ((2/3) ^ m - (2/3) ^ n)"
 by (simp add: sum_distrib_left 23)
 also have "... < (c/3) * 3 * ((2/3) ^ m)"
 using \<open>0 < c\<close> by auto
 also have "\ < \"
 using N [OF \<open>N \<le> m\<close>] \<open>0 < c\<close> by (simp add: field_simps)
 finally show ?thesis .
 qed (use \<open>0 < \<epsilon>\<close> \<open>m \<le> n\<close> in auto)
 then show ?thesis
 by (metis dist_commute_lessI dist_real_def nle_le)
 qed
 qed auto
 define \<phi> where "\<phi> \<equiv> \<lambda>x. max a (min (h x) b)"
 show thesis
 proof
 show "continuous_map X euclidean \"
 unfolding \<phi>_def using conth by (intro continuous_intros) auto
 show "\ x = f x" if "x \ S" for x
 proof -
 have x: "x \ topspace X"
 using \<open>closedin X S\<close> closedin_subset that by blast
 have "h x = f x"
 proof (rule Met_TC.limitin_metric_unique)
 show "limitin Met_TC.mtopology (\n. g n x) (h x) sequentially"
 using h x by (force simp: tendsto_iff eventually_sequentially)
 show "limitin Met_TC.mtopology (\n. g n x) (f x) sequentially"
 proof (clarsimp simp: tendsto_iff)
 fix \<epsilon>::real
 assume "\ > 0"
 then have "\\<^sub>F n in sequentially. \(2/3) ^ n\ < \/c"
 by (intro Archimedean_eventually_pow_inverse) (auto simp: \<open>c > 0\<close>)
 then show "\\<^sub>F n in sequentially. dist (g n x) (f x) < \"
 apply eventually_elim
 by (smt (verit) good x good_def \<open>c > 0\<close> dist_real_def mult.commute pos_less_divide_eq that)
 qed
 qed auto
 then show ?thesis
 using that fim by (auto simp: \<phi>_def)
 qed
 then show "\ ` topspace X \ {a..b}"
 using fim \<open>a \<le> b\<close> by (auto simp: \<phi>_def)
 qed
qed

theorem Tietze_extension_realinterval:
 assumes XS: "normal_space X" "closedin X S" and T: "is_interval T" "T \ {}"
 and contf: "continuous_map (subtopology X S) euclideanreal f"
 and "f ` S \ T"
 obtains g where "continuous_map X euclideanreal g" "g ` topspace X \ T" "\x. x \ S \ g x = f x"
proof -
 define \<Phi> where
 "\ \ \T::real set. \f. continuous_map (subtopology X S) euclidean f \ f`S \ T
 \<longrightarrow> (\<exists>g. continuous_map X euclidean g \<and> g ` topspace X \<subseteq> T \<and> (\<forall>x \<in> S. g x = f x))"
 have "\ T"
 if *: "\T. \bounded T; is_interval T; T \ {}\ \ \ T"
 and "is_interval T" "T \ {}" for T
 unfolding \<Phi>_def
 proof (intro strip)
 fix f
 assume contf: "continuous_map (subtopology X S) euclideanreal f"
 and "f ` S \ T"
 have \<Phi>T: "\<Phi> ((\<lambda>x. x / (1 + \<bar>x\<bar>)) ` T)"
 proof (rule *)
 show "bounded ((\x. x / (1 + \x\)) ` T)"
 using shrink_range [of T] by (force intro: boundedI [where B=1])
 show "is_interval ((\x. x / (1 + \x\)) ` T)"
 using connected_shrink that(2) is_interval_connected_1 by blast
 show "(\x. x / (1 + \x\)) ` T \ {}"
 using \<open>T \<noteq> {}\<close> by auto
 qed
 moreover have "continuous_map (subtopology X S) euclidean ((\x. x / (1 + \x\)) \ f)"
 by (metis contf continuous_map_compose continuous_map_into_fulltopology continuous_map_real_shrink)
 moreover have "((\x. x / (1 + \x\)) \ f) ` S \ (\x. x / (1 + \x\)) ` T"
 using \<open>f ` S \<subseteq> T\<close> by auto
 ultimately obtain g
 where contg: "continuous_map X euclidean g"
 and gim: "g ` topspace X \ (\x. x / (1 + \x\)) ` T"
 and geq: "\x. x \ S \ g x = ((\x. x / (1 + \x\)) \ f) x"
 using \<Phi>T unfolding \<Phi>_def by force
 show "\g. continuous_map X euclideanreal g \ g ` topspace X \ T \ (\x\S. g x = f x)"
 proof (intro conjI exI)
 have "continuous_map X (top_of_set {-1<..<1}) g"
 using contg continuous_map_in_subtopology gim shrink_range by blast
 then show "continuous_map X euclideanreal ((\x. x / (1 - \x\)) \ g)"
 by (rule continuous_map_compose) (auto simp: continuous_on_real_grow)
 show "((\x. x / (1 - \x\)) \ g) ` topspace X \ T"
 using gim real_grow_shrink by fastforce
 show "\x\S. ((\x. x / (1 - \x\)) \ g) x = f x"
 using geq real_grow_shrink by force
 qed
 qed
 moreover have "\ T"
 if "bounded T" "is_interval T" "T \ {}" for T
 unfolding \<Phi>_def
 proof (intro strip)
 fix f
 assume contf: "continuous_map (subtopology X S) euclideanreal f"
 and "f ` S \ T"
 obtain a b where ab: "closure T = {a..b}"
 by (meson \<open>bounded T\<close> \<open>is_interval T\<close> compact_closure connected_compact_interval_1
 connected_imp_connected_closure is_interval_connected)
 with \<open>T \<noteq> {}\<close> have "a \<le> b" by auto
 have "f ` S \ {a..b}"
 using \<open>f ` S \<subseteq> T\<close> ab closure_subset by auto
 then obtain g where contg: "continuous_map X euclideanreal g"
 and gf: "\x. x \ S \ g x = f x" and gim: "g ` topspace X \ {a..b}"
 using Tietze_extension_closed_real_interval [OF XS contf _ \<open>a \<le> b\<close>] by metis
 define W where "W \ {x \ topspace X. g x \ closure T - T}"
 have "{a..b} - {a, b} \ T"
 using that
 by (metis ab atLeastAtMost_diff_ends convex_interior_closure interior_atLeastAtMost_real
 interior_subset is_interval_convex)
 with finite_imp_compact have "compact (closure T - T)"
 by (metis Diff_eq_empty_iff Diff_insert2 ab finite.emptyI finite_Diff_insert)
 then have "closedin X W"
 unfolding W_def using closedin_continuous_map_preimage [OF contg] compact_imp_closed by force
 moreover have "disjnt W S"
 unfolding W_def disjnt_iff using \<open>f ` S \<subseteq> T\<close> gf by blast
 ultimately obtain h :: "'a \ real"
 where conth: "continuous_map X (top_of_set {0..1}) h"
 and him: "h ` W \ {0}" "h ` S \ {1}"
 by (metis XS normal_space_iff_Urysohn)
 then have him01: "h \ topspace X \ {0..1}"
 by (metis continuous_map_in_subtopology)
 obtain z where "z \ T"
 using \<open>T \<noteq> {}\<close> by blast
 define g' where "g' \<equiv> \<lambda>x. z + h x * (g x - z)"
 show "\g. continuous_map X euclidean g \ g ` topspace X \ T \ (\x\S. g x = f x)"
 proof (intro exI conjI)
 show "continuous_map X euclideanreal g'"
 unfolding g'_def using contg conth continuous_map_in_subtopology
 by (intro continuous_intros) auto
 show "g' ` topspace X \ T"
 unfolding g'_def
 proof clarify
 fix x
 assume "x \ topspace X"
 show "z + h x * (g x - z) \ T"
 proof (cases "g x \ T")
 case True
 define w where "w \ z + h x * (g x - z)"
 have "\h x\ * \g x - z\ \ \g x - z\" "\1 - h x\ * \g x - z\ \ \g x - z\"
 using him01 \<open>x \<in> topspace X\<close> by (force simp: intro: mult_left_le_one_le)+
 then consider "z \ w \ w \ g x" | "g x \ w \ w \ z"
 unfolding w_def by (smt (verit) left_diff_distrib mult_cancel_right2 mult_minus_right zero_less_mult_iff)
 then show ?thesis
 using \<open>is_interval T\<close> unfolding w_def is_interval_1 by (metis True \<open>z \<in> T\<close>)
 next
 case False
 then have "g x \ closure T"
 using \<open>x \<in> topspace X\<close> ab gim by blast
 then have "h x = 0"
 using him False \<open>x \<in> topspace X\<close> by (auto simp: W_def image_subset_iff)
 then show ?thesis
 by (simp add: \<open>z \<in> T\<close>)
 qed
 qed
 show "\x\S. g' x = f x"
 using gf him by (auto simp: W_def g'_def)
 qed
 qed
 ultimately show thesis
 using assms that unfolding \<Phi>_def by best
qed

lemma normal_space_iff_Tietze:
 "normal_space X \
 (\<forall>f S. closedin X S \<and>
 continuous_map (subtopology X S) euclidean f
 \<longrightarrow> (\<exists>g:: 'a \<Rightarrow> real. continuous_map X euclidean g \<and> (\<forall>x \<in> S. g x = f x)))"
 (is "?lhs \ ?rhs")
proof
 assume ?lhs
 then show ?rhs
 by (metis Tietze_extension_realinterval empty_not_UNIV is_interval_univ subset_UNIV)
next
 assume R: ?rhs
 show ?lhs
 unfolding normal_space_iff_Urysohn_alt
 proof clarify
 fix S T
 assume "closedin X S"
 and "closedin X T"
 and "disjnt S T"
 then have cloST: "closedin X (S \ T)"
 by (simp add: closedin_Un)
 moreover
 have "continuous_map (subtopology X (S \ T)) euclideanreal (\x. if x \ S then 0 else 1)"
 unfolding continuous_map_closedin
 proof (intro conjI strip)
 fix C :: "real set"
 define D where "D \ {x \ topspace X. if x \ S then 0 \ C else x \ T \ 1 \ C}"
 have "D \ {{}, S, T, S \ T}"
 unfolding D_def
 using closedin_subset [OF \<open>closedin X S\<close>] closedin_subset [OF \<open>closedin X T\<close>] \<open>disjnt S T\<close>
 by (auto simp: disjnt_iff)
 then have "closedin X D"
 using \<open>closedin X S\<close> \<open>closedin X T\<close> closedin_empty by blast
 with closedin_subset_topspace
 show "closedin (subtopology X (S \ T)) {x \ topspace (subtopology X (S \ T)). (if x \ S then 0::real else 1) \ C}"
 apply (simp add: D_def)
 by (smt (verit, best) Collect_cong Collect_mono_iff Un_def closedin_subset_topspace)
 qed auto
 ultimately obtain g :: "'a \ real" where
 contg: "continuous_map X euclidean g" and gf: "\x \ S \ T. g x = (if x \ S then 0 else 1)"
 using R by blast
 then show "\f. continuous_map X euclideanreal f \ f ` S \ {0} \ f ` T \ {1}"
 by (smt (verit) Un_iff \<open>disjnt S T\<close> disjnt_iff image_subset_iff insert_iff)
 qed
qed

subsection \<open>Random metric space stuff\<close>

lemma metrizable_imp_k_space:
 assumes "metrizable_space X"
 shows "k_space X"
proof -
 obtain M d where "Metric_space M d" and Xeq: "X = Metric_space.mtopology M d"
 using assms unfolding metrizable_space_def by metis
 then interpret Metric_space M d
 by blast
 show ?thesis
 unfolding k_space Xeq
 proof clarsimp
 fix S
 assume "S \ M" and S: "\K. compactin mtopology K \ closedin (subtopology mtopology K) (K \ S)"
 have "l \ S"
 if \<sigma>: "range \<sigma> \<subseteq> S" and l: "limitin mtopology \<sigma> l sequentially" for \<sigma> l
 proof -
 define K where "K \ insert l (range \)"
 interpret Submetric M d K
 proof
 show "K \ M"
 unfolding K_def using l limitin_mspace \<open>S \<subseteq> M\<close> \<sigma> by blast
 qed
 have "compactin mtopology K"
 unfolding K_def using \<open>S \<subseteq> M\<close> \<sigma>
 by (force intro: compactin_sequence_with_limit [OF l])
 then have *: "closedin sub.mtopology (K \ S)"
 by (simp add: S mtopology_submetric)
 have "\ n \ K \ S" for n
 by (simp add: K_def range_subsetD \<sigma>)
 moreover have "limitin sub.mtopology \ l sequentially"
 using l
 unfolding sub.limit_metric_sequentially limit_metric_sequentially
 by (force simp: K_def)
 ultimately have "l \ K \ S"
 by (meson * \<sigma> image_subsetI sub.metric_closedin_iff_sequentially_closed)
 then show ?thesis
 by simp
 qed
 then show "closedin mtopology S"
 unfolding metric_closedin_iff_sequentially_closed
 using \<open>S \<subseteq> M\<close> by blast
 qed
qed

lemma (in Metric_space) k_space_mtopology: "k_space mtopology"
 by (simp add: metrizable_imp_k_space metrizable_space_mtopology)

lemma k_space_euclideanreal: "k_space (euclidean :: 'a::metric_space topology)"
 using metrizable_imp_k_space metrizable_space_euclidean by auto

subsection\<open>Hereditarily normal spaces\<close>

lemma hereditarily_B:
 assumes "\S T. separatedin X S T
 \<Longrightarrow> \<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V"
 shows "hereditarily normal_space X"
 unfolding hereditarily_def
proof (intro strip)
 fix W
 assume "W \ topspace X"
 show "normal_space (subtopology X W)"
 unfolding normal_space_def
 proof clarify
 fix S T
 assume clo: "closedin (subtopology X W) S" "closedin (subtopology X W) T"
 and "disjnt S T"
 then have "separatedin (subtopology X W) S T"
 by (simp add: separatedin_closed_sets)
 then obtain U V where "openin X U \ openin X V \ S \ U \ T \ V \ disjnt U V"
 using assms [of S T] by (meson separatedin_subtopology)
 then show "\U V. openin (subtopology X W) U \ openin (subtopology X W) V \ S \ U \ T \ V \ disjnt U V"
 apply (simp add: openin_subtopology_alt)
 by (meson clo closedin_imp_subset disjnt_subset1 disjnt_subset2 inf_le2)
 qed
qed

lemma hereditarily_C:
 assumes "separatedin X S T" and norm: "\U. openin X U \ normal_space (subtopology X U)"
 shows "\U V. openin X U \ openin X V \ S \ U \ T \ V \ disjnt U V"
proof -
 define ST where "ST \ X closure_of S \ X closure_of T"
 have subX: "S \ topspace X" "T \ topspace X"
 by (meson \<open>separatedin X S T\<close> separation_closedin_Un_gen)+
 have sub: "S \ topspace X - ST" "T \ topspace X - ST"
 unfolding ST_def
 by (metis Diff_mono Diff_triv \<open>separatedin X S T\<close> Int_lower1 Int_lower2 separatedin_def)+
 have "normal_space (subtopology X (topspace X - ST))"
 by (simp add: ST_def norm closedin_Int openin_diff)
 moreover have " disjnt (subtopology X (topspace X - ST) closure_of S)
 (subtopology X (topspace X - ST) closure_of T)"
 using Int_absorb1 ST_def sub by (fastforce simp: disjnt_iff closure_of_subtopology)
 ultimately show ?thesis
 using sub subX
 apply (simp add: normal_space_closures)
 by (metis ST_def closedin_Int closedin_closure_of closedin_def openin_trans_full)
qed

lemma hereditarily_normal_space:
 "hereditarily normal_space X \ (\U. openin X U \ normal_space(subtopology X U))"
 by (metis hereditarily_B hereditarily_C hereditarily)

lemma hereditarily_normal_separation:
 "hereditarily normal_space X \
 (\<forall>S T. separatedin X S T
 \<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V))"
 by (metis hereditarily_B hereditarily_C hereditarily)

lemma metrizable_imp_hereditarily_normal_space:
 "metrizable_space X \ hereditarily normal_space X"
 by (simp add: hereditarily metrizable_imp_normal_space metrizable_space_subtopology)

lemma metrizable_space_separation:
 "\metrizable_space X; separatedin X S T\
 \<Longrightarrow> \<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V"
 by (metis hereditarily hereditarily_C metrizable_imp_hereditarily_normal_space)

lemma hereditarily_normal_separation_pairwise:
 "hereditarily normal_space X \
 (\<forall>\. finite \ \<and> (\<forall>S \<in> \. S \<subseteq> topspace X) \<and> pairwise (separatedin X) \
 \<longrightarrow> (\<exists>\<F>. (\<forall>S \<in> \. openin X (\<F> S) \<and> S \<subseteq> \<F> S) \<and>
 pairwise (\<lambda>S T. disjnt (\<F> S) (\<F> T)) \))"
 (is "?lhs \ ?rhs")
proof
 assume L: ?lhs
 show ?rhs
 proof clarify
 fix \
 assume "finite \" and \: "\S\\. S \ topspace X"
 and pw\: "pairwise (separatedin X) \"
 have "\V W. openin X V \ openin X W \ S \ V \
 (\<forall>T. T \<in> \ \<and> T \<noteq> S \<longrightarrow> T \<subseteq> W) \<and> disjnt V W"
 if "S \ \" for S
 proof -
 have "separatedin X S (\(\ - {S}))"
 by (metis \ \<open>finite \\<close> pw\ finite_Diff pairwise_alt separatedin_Union(1) that)
 with L show ?thesis
 unfolding hereditarily_normal_separation
 by (smt (verit) Diff_iff UnionI empty_iff insert_iff subset_iff)
 qed
 then obtain \<F> \<G>
 where *: "\S. S \ \ \ S \ \ S \ (\T. T \ \ \ T \ S \ T \ \ S)"
 and ope: "\S. S \ \ \ openin X (\ S) \ openin X (\ S)"
 and dis: "\S. S \ \ \ disjnt (\ S) (\ S)"
 by metis
 define \<H> where "\<H> \<equiv> \<lambda>S. \<F> S \<inter> (\<Inter>T \<in> \ - {S}. \<G> T)"
 show "\\. (\S\\. openin X (\ S) \ S \ \ S) \ pairwise (\S T. disjnt (\ S) (\ T)) \"
 proof (intro exI conjI strip)
 show "openin X (\ S)" if "S \ \" for S
 unfolding \<H>_def
 by (smt (verit) ope that DiffD1 \<open>finite \\<close> finite_Diff finite_imageI imageE openin_Int_Inter)
 show "S \ \ S" if "S \ \" for S
 unfolding \<H>_def using "*" that by auto
 show "pairwise (\S T. disjnt (\ S) (\ T)) \"
 using dis by (fastforce simp: disjnt_iff pairwise_alt \<H>_def)
 qed
 qed
next
 assume R: ?rhs
 show ?lhs
 unfolding hereditarily_normal_separation
 proof (intro strip)
 fix S T
 assume "separatedin X S T"
 show "\U V. openin X U \ openin X V \ S \ U \ T \ V \ disjnt U V"
 proof (cases "T=S")
 case True
 then show ?thesis
 using \<open>separatedin X S T\<close> by force
 next
 case False
 have "pairwise (separatedin X) {S, T}"
 by (simp add: \<open>separatedin X S T\<close> pairwise_insert separatedin_sym)
 moreover have "\S\{S, T}. S \ topspace X"
 by (metis \<open>separatedin X S T\<close> insertE separatedin_def singletonD)
 ultimately show ?thesis
 using R by (smt (verit) False finite.emptyI finite.insertI insertCI pairwiseD)
 qed
 qed
qed

lemma hereditarily_normal_space_perfect_map_image:
 "\hereditarily normal_space X; perfect_map X Y f\ \ hereditarily normal_space Y"
 unfolding perfect_map_def proper_map_def
 by (meson hereditarily_normal_space_continuous_closed_map_image)

lemma regular_second_countable_imp_hereditarily_normal_space:
 assumes "regular_space X \ second_countable X"
 shows "hereditarily normal_space X"
 unfolding hereditarily
 proof (intro regular_Lindelof_imp_normal_space strip)
 show "regular_space (subtopology X S)" for S
 by (simp add: assms regular_space_subtopology)
 show "Lindelof_space (subtopology X S)" for S
 using assms by (simp add: second_countable_imp_Lindelof_space second_countable_subtopology)
qed

subsection\<open>Completely regular spaces\<close>

definition completely_regular_space where
"completely_regular_space X \
 \<forall>S x. closedin X S \<and> x \<in> topspace X - S
 \<longrightarrow> (\<exists>f::'a\<Rightarrow>real. continuous_map X (top_of_set {0..1}) f \<and>
 f x = 0 \<and> (f ` S \<subseteq> {1}))"

lemma homeomorphic_completely_regular_space_aux:
 assumes X: "completely_regular_space X" and hom: "X homeomorphic_space Y"
 shows "completely_regular_space Y"
proof -
 obtain f g where hmf: "homeomorphic_map X Y f" and hmg: "homeomorphic_map Y X g"
 and fg: "(\x \ topspace X. g(f x) = x) \ (\y \ topspace Y. f(g y) = y)"
 using assms X homeomorphic_maps_map homeomorphic_space_def by fastforce
 show ?thesis
 unfolding completely_regular_space_def
 proof clarify
 fix S x
 assume A: "closedin Y S" and x: "x \ topspace Y" and "x \ S"
 then have "closedin X (g`S)"
 using hmg homeomorphic_map_closedness_eq by blast
 moreover have "g x \ g`S"
 by (meson A x \<open>x \<notin> S\<close> closedin_subset hmg homeomorphic_imp_injective_map inj_on_image_mem_iff)
 ultimately obtain \<phi> where \<phi>: "continuous_map X (top_of_set {0..1::real}) \<phi> \<and> \<phi> (g x) = 0 \<and> \<phi> ` g`S \<subseteq> {1}"
 by (metis DiffI X completely_regular_space_def hmg homeomorphic_imp_surjective_map image_eqI x)
 then have "continuous_map Y (top_of_set {0..1::real}) (\ \ g)"
 by (meson continuous_map_compose hmg homeomorphic_imp_continuous_map)
 then show "\\. continuous_map Y (top_of_set {0..1::real}) \ \ \ x = 0 \ \ ` S \ {1}"
 by (metis \<phi> comp_apply image_comp)
 qed
qed

lemma homeomorphic_completely_regular_space:
 assumes "X homeomorphic_space Y"
 shows "completely_regular_space X \ completely_regular_space Y"
 by (meson assms homeomorphic_completely_regular_space_aux homeomorphic_space_sym)

lemma completely_regular_space_alt:
 "completely_regular_space X \
 (\<forall>S x. closedin X S \<longrightarrow> x \<in> topspace X - S
 \<longrightarrow> (\<exists>f. continuous_map X euclideanreal f \<and> f x = 0 \<and> f ` S \<subseteq> {1}))"
proof -
 have "\f. continuous_map X (top_of_set {0..1::real}) f \ f x = 0 \ f ` S \ {1}"
 if "closedin X S" "x \ topspace X - S" and f: "continuous_map X euclideanreal f \ f x = 0 \ f ` S \ {1}"
 for S x f
 proof (intro exI conjI)
 show "continuous_map X (top_of_set {0..1}) (\x. max 0 (min (f x) 1))"
 using that
 by (auto simp: continuous_map_in_subtopology intro!: continuous_map_real_max continuous_map_real_min)
 qed (use that in auto)
 with continuous_map_in_subtopology show ?thesis
 unfolding completely_regular_space_def by metis
qed

text \<open>As above, but with @{term openin}\<close>
lemma completely_regular_space_alt':
 "completely_regular_space X \
 (\<forall>S x. openin X S \<longrightarrow> x \<in> S
 \<longrightarrow> (\<exists>f. continuous_map X euclideanreal f \<and> f x = 0 \<and> f ` (topspace X - S) \<subseteq> {1}))"
 apply (simp add: completely_regular_space_alt all_closedin)
 by (meson openin_subset subsetD)

lemma completely_regular_space_gen_alt:
 fixes a b::real
 assumes "a \ b"
 shows "completely_regular_space X \
 (\<forall>S x. closedin X S \<longrightarrow> x \<in> topspace X - S
 \<longrightarrow> (\<exists>f. continuous_map X euclidean f \<and> f x = a \<and> (f ` S \<subseteq> {b})))"
proof -
 have "\f. continuous_map X euclideanreal f \ f x = 0 \ f ` S \ {1}"
 if "closedin X S" "x \ topspace X - S"
 and f: "continuous_map X euclidean f \ f x = a \ f ` S \ {b}"
 for S x f
 proof (intro exI conjI)
 show "continuous_map X euclideanreal ((\x. inverse(b - a) * (x - a)) \ f)"
 using that by (intro continuous_intros) auto
 qed (use that assms in auto)
 moreover
 have "\f. continuous_map X euclidean f \ f x = a \ f ` S \ {b}"
 if "closedin X S" "x \ topspace X - S"
 and f: "continuous_map X euclideanreal f \ f x = 0 \ f ` S \ {1}"
 for S x f
 proof (intro exI conjI)
 show "continuous_map X euclideanreal ((\x. a + (b - a) * x) \ f)"
 using that by (intro continuous_intros) auto
 qed (use that in auto)
 ultimately show ?thesis
 unfolding completely_regular_space_alt by meson
qed

text \<open>As above, but with @{term openin}\<close>
lemma completely_regular_space_gen_alt':
 fixes a b::real
 assumes "a \ b"
 shows "completely_regular_space X \
 (\<forall>S x. openin X S \<longrightarrow> x \<in> S
 \<longrightarrow> (\<exists>f. continuous_map X euclidean f \<and> f x = a \<and> (f ` (topspace X - S) \<subseteq> {b})))"
 apply (simp add: completely_regular_space_gen_alt[OF assms] all_closedin)
 by (meson openin_subset subsetD)

lemma completely_regular_space_gen:
 fixes a b::real
 assumes "a < b"
 shows "completely_regular_space X \
 (\<forall>S x. closedin X S \<and> x \<in> topspace X - S
 \<longrightarrow> (\<exists>f. continuous_map X (top_of_set {a..b}) f \<and> f x = a \<and> f ` S \<subseteq> {b}))"
proof -
 have "\f. continuous_map X (top_of_set {a..b}) f \ f x = a \ f ` S \ {b}"
 if "closedin X S" "x \ topspace X - S"
 and f: "continuous_map X euclidean f \ f x = a \ f ` S \ {b}"
 for S x f
 proof (intro exI conjI)
 show "continuous_map X (top_of_set {a..b}) (\x. max a (min (f x) b))"
 using that assms
 by (auto simp: continuous_map_in_subtopology intro!: continuous_map_real_max continuous_map_real_min)
 qed (use that assms in auto)
 with continuous_map_in_subtopology assms show ?thesis
 using completely_regular_space_gen_alt [of a b]
 by (smt (verit) atLeastAtMost_singleton atLeastatMost_empty singletonI)
qed

lemma normal_imp_completely_regular_space_A:
 assumes "normal_space X" "t1_space X"
 shows "completely_regular_space X"
 unfolding completely_regular_space_alt
proof clarify
 fix x S
 assume A: "closedin X S" "x \ S"
 assume "x \ topspace X"
 then have "closedin X {x}"
 by (simp add: \<open>t1_space X\<close> closedin_t1_singleton)
 with A \<open>normal_space X\<close> have "\<exists>f. continuous_map X euclideanreal f \<and> f ` {x} \<subseteq> {0} \<and> f ` S \<subseteq> {1}"
 using assms unfolding normal_space_iff_Urysohn_alt disjnt_iff by blast
 then show "\f. continuous_map X euclideanreal f \ f x = 0 \ f ` S \ {1}"
 by auto
qed

lemma normal_imp_completely_regular_space_B:
 assumes "normal_space X" "regular_space X"
 shows "completely_regular_space X"
 unfolding completely_regular_space_alt
proof clarify
 fix x S
 assume "closedin X S" "x \ S" "x \ topspace X"
 then obtain U C where "openin X U" "closedin X C" "x \ U" "U \ C" "C \ topspace X - S"
 using assms
 unfolding neighbourhood_base_of_closedin [symmetric] neighbourhood_base_of closedin_def by (metis Diff_iff)
 then obtain f where "continuous_map X euclideanreal f \ f ` C \ {0} \ f ` S \ {1}"
 using assms unfolding normal_space_iff_Urysohn_alt
 by (metis Diff_iff \<open>closedin X S\<close> disjnt_iff subsetD)
 then show "\f. continuous_map X euclideanreal f \ f x = 0 \ f ` S \ {1}"
 by (meson \<open>U \<subseteq> C\<close> \<open>x \<in> U\<close> image_subset_iff singletonD subsetD)
qed

lemma normal_imp_completely_regular_space_gen:
 "\normal_space X; t1_space X \ Hausdorff_space X \ regular_space X\ \ completely_regular_space X"
 using normal_imp_completely_regular_space_A normal_imp_completely_regular_space_B t1_or_Hausdorff_space by blast

lemma normal_imp_completely_regular_space:
 "\normal_space X; Hausdorff_space X \ regular_space X\ \ completely_regular_space X"
 by (simp add: normal_imp_completely_regular_space_gen)

lemma (in Metric_space) completely_regular_space_mtopology:
 "completely_regular_space mtopology"
 by (simp add: normal_imp_completely_regular_space normal_space_mtopology regular_space_mtopology)

lemma metrizable_imp_completely_regular_space:
 "metrizable_space X \ completely_regular_space X"
 by (simp add: metrizable_imp_normal_space metrizable_imp_regular_space normal_imp_completely_regular_space)

lemma completely_regular_space_discrete_topology:
 "completely_regular_space(discrete_topology U)"
 by (simp add: normal_imp_completely_regular_space normal_space_discrete_topology)

lemma completely_regular_space_subtopology:
 assumes "completely_regular_space X"
 shows "completely_regular_space (subtopology X S)"
 unfolding completely_regular_space_def
proof clarify
 fix A x
 assume "closedin (subtopology X S) A" and x: "x \ topspace (subtopology X S)" and "x \ A"
 then obtain T where "closedin X T" "A = S \ T" "x \ topspace X" "x \ S"
 by (force simp: closedin_subtopology_alt image_iff)
 then show "\f. continuous_map (subtopology X S) (top_of_set {0::real..1}) f \ f x = 0 \ f ` A \ {1}"
 using assms \<open>x \<notin> A\<close>
 apply (simp add: completely_regular_space_def continuous_map_from_subtopology)
 using continuous_map_from_subtopology by fastforce
qed

lemma completely_regular_space_retraction_map_image:
 " \retraction_map X Y r; completely_regular_space X\ \ completely_regular_space Y"
 using completely_regular_space_subtopology hereditary_imp_retractive_property homeomorphic_completely_regular_space by blast

lemma completely_regular_imp_regular_space:
 assumes "completely_regular_space X"
 shows "regular_space X"
proof -
 have *: "\U V. openin X U \ openin X V \ a \ U \ C \ V \ disjnt U V"
 if contf: "continuous_map X euclideanreal f" and a: "a \ topspace X - C" and "closedin X C"
 and fim: "f \ topspace X \ {0..1}" and f0: "f a = 0" and f1: "f ` C \ {1}"
 for C a f
 proof (intro exI conjI)
 show "openin X {x \ topspace X. f x \ {..<1 / 2}}" "openin X {x \ topspace X. f x \ {1 / 2<..}}"
 using openin_continuous_map_preimage [OF contf]
 by (meson open_lessThan open_greaterThan open_openin)+
 show "a \ {x \ topspace X. f x \ {..<1 / 2}}"
 using a f0 by auto
 show "C \ {x \ topspace X. f x \ {1 / 2<..}}"
 using \<open>closedin X C\<close> f1 closedin_subset by auto
 qed (auto simp: disjnt_iff)
 show ?thesis
 using assms *
 unfolding completely_regular_space_def regular_space_def continuous_map_in_subtopology
 by metis
qed

proposition locally_compact_regular_imp_completely_regular_space:
 assumes "locally_compact_space X" "Hausdorff_space X \ regular_space X"
 shows "completely_regular_space X"
 unfolding completely_regular_space_def
proof clarify
 fix S x
 assume "closedin X S" and "x \ topspace X" and "x \ S"
 have "regular_space X"
 using assms locally_compact_Hausdorff_imp_regular_space by blast
 then have nbase: "neighbourhood_base_of (\C. compactin X C \ closedin X C) X"
 using assms(1) locally_compact_regular_space_neighbourhood_base by blast
 then obtain U M where "openin X U" "compactin X M" "closedin X M" "x \ U" "U \ M" "M \ topspace X - S"
 unfolding neighbourhood_base_of by (metis (no_types, lifting) Diff_iff \<open>closedin X S\<close> \<open>x \<in> topspace X\<close> \<open>x \<notin> S\<close> closedin_def)
 then have "M \ topspace X"
 by blast
 obtain V K where "openin X V" "closedin X K" "x \ V" "V \ K" "K \ U"
 by (metis (no_types, lifting) \<open>openin X U\<close> \<open>x \<in> U\<close> neighbourhood_base_of nbase)
 have "compact_space (subtopology X M)"
 by (simp add: \<open>compactin X M\<close> compact_space_subtopology)
 then have "normal_space (subtopology X M)"
 by (simp add: \<open>regular_space X\<close> compact_Hausdorff_or_regular_imp_normal_space regular_space_subtopology)
 moreover have "closedin (subtopology X M) K"
 using \<open>K \<subseteq> U\<close> \<open>U \<subseteq> M\<close> \<open>closedin X K\<close> closedin_subset_topspace by fastforce
 moreover have "closedin (subtopology X M) (M - U)"
 by (simp add: \<open>closedin X M\<close> \<open>openin X U\<close> closedin_diff closedin_subset_topspace)
 moreover have "disjnt K (M - U)"
 by (meson DiffD2 \<open>K \<subseteq> U\<close> disjnt_iff subsetD)
 ultimately obtain f::"'a\real" where contf: "continuous_map (subtopology X M) (top_of_set {0..1}) f"
 and f0: "f ` K \ {0}" and f1: "f ` (M - U) \ {1}"
 using Urysohn_lemma [of "subtopology X M" K "M-U" 0 1] by auto
 then obtain g::"'a\real" where contg: "continuous_map (subtopology X M) euclidean g" and gim: "g ` M \ {0..1}"
 and g0: "\x. x \ K \ g x = 0" and g1: "\x. \x \ M; x \ U\ \ g x = 1"
 using \<open>M \<subseteq> topspace X\<close> by (force simp: continuous_map_in_subtopology image_subset_iff)
 show "\f::'a\real. continuous_map X (top_of_set {0..1}) f \ f x = 0 \ f ` S \ {1}"
 proof (intro exI conjI)
 show "continuous_map X (top_of_set {0..1}) (\x. if x \ M then g x else 1)"
 unfolding continuous_map_closedin
 proof (intro strip conjI)
 fix C
 assume C: "closedin (top_of_set {0::real..1}) C"
 have eq: "{x \ topspace X. (if x \ M then g x else 1) \ C} = {x \ M. g x \ C} \ (if 1 \ C then topspace X - U else {})"
 using \<open>U \<subseteq> M\<close> \<open>M \<subseteq> topspace X\<close> g1 by auto
 show "closedin X {x \ topspace X. (if x \ M then g x else 1) \ C}"
 unfolding eq
 proof (intro closedin_Un)
 have "closedin euclidean C"
 using C closed_closedin closedin_closed_trans by blast
 then have "closedin (subtopology X M) {x \ M. g x \ C}"
 using closedin_continuous_map_preimage_gen [OF contg] \<open>M \<subseteq> topspace X\<close> by auto
 then show "closedin X {x \ M. g x \ C}"
 using \<open>closedin X M\<close> closedin_trans_full by blast
 qed (use \<open>openin X U\<close> in force)
 qed (use gim in force)
 show "(if x \ M then g x else 1) = 0"
 using \<open>U \<subseteq> M\<close> \<open>V \<subseteq> K\<close> g0 \<open>x \<in> U\<close> \<open>x \<in> V\<close> by auto
 show "(\x. if x \ M then g x else 1) ` S \ {1}"
 using \<open>M \<subseteq> topspace X - S\<close> by auto
 qed
qed

lemma completely_regular_eq_regular_space:
 "locally_compact_space X
 \<Longrightarrow> (completely_regular_space X \<longleftrightarrow> regular_space X)"
 using completely_regular_imp_regular_space locally_compact_regular_imp_completely_regular_space
 by blast

lemma completely_regular_space_prod_topology:
 "completely_regular_space (prod_topology X Y) \
 (prod_topology X Y) = trivial_topology \<or>
 completely_regular_space X \<and> completely_regular_space Y" (is "?lhs=?rhs")
proof
 assume ?lhs then show ?rhs
 by (rule topological_property_of_prod_component)
 (auto simp: completely_regular_space_subtopology homeomorphic_completely_regular_space)
next
 assume R: ?rhs
 show ?lhs
 proof (cases "(prod_topology X Y) = trivial_topology")
 case False
 then have X: "completely_regular_space X" and Y: "completely_regular_space Y"
 using R by blast+
 show ?thesis
 unfolding completely_regular_space_alt'
 proof clarify
 fix W x y
 assume "openin (prod_topology X Y) W" and "(x, y) \ W"
 then obtain U V where "openin X U" "openin Y V" "x \ U" "y \ V" "U\V \ W"
 by (force simp: openin_prod_topology_alt)
 then have "x \ topspace X" "y \ topspace Y"
 using openin_subset by fastforce+
 obtain f where contf: "continuous_map X euclideanreal f" and "f x = 0"
 and f1: "\x. x \ topspace X \ x \ U \ f x = 1"
 using X \<open>openin X U\<close> \<open>x \<in> U\<close> unfolding completely_regular_space_alt'
 by (smt (verit, best) Diff_iff image_subset_iff singletonD)
 obtain g where contg: "continuous_map Y euclideanreal g" and "g y = 0"
 and g1: "\y. y \ topspace Y \ y \ V \ g y = 1"
 using Y \<open>openin Y V\<close> \<open>y \<in> V\<close> unfolding completely_regular_space_alt'
 by (smt (verit, best) Diff_iff image_subset_iff singletonD)
 define h where "h \ \(x,y). 1 - (1 - f x) * (1 - g y)"
 show "\h. continuous_map (prod_topology X Y) euclideanreal h \ h (x,y) = 0 \ h ` (topspace (prod_topology X Y) - W) \ {1}"
 proof (intro exI conjI)
 have "continuous_map (prod_topology X Y) euclideanreal (f \ fst)"
 using contf continuous_map_of_fst by blast
 moreover
 have "continuous_map (prod_topology X Y) euclideanreal (g \ snd)"
 using contg continuous_map_of_snd by blast
 ultimately
 show "continuous_map (prod_topology X Y) euclideanreal h"
 unfolding o_def h_def case_prod_unfold
 by (intro continuous_intros) auto
 show "h (x, y) = 0"
 by (simp add: h_def \<open>f x = 0\<close> \<open>g y = 0\<close>)
 show "h ` (topspace (prod_topology X Y) - W) \ {1}"
 using \<open>U \<times> V \<subseteq> W\<close> f1 g1 by (force simp: h_def)
 qed
 qed
 qed (force simp: completely_regular_space_def)
qed

proposition completely_regular_space_product_topology:
 "completely_regular_space (product_topology X I) \
 (\<exists>i\<in>I. X i = trivial_topology) \<or> (\<forall>i \<in> I. completely_regular_space (X i))"
 (is "?lhs \ ?rhs")
proof
 assume ?lhs then show ?rhs
 by (rule topological_property_of_product_component)
 (auto simp: completely_regular_space_subtopology homeomorphic_completely_regular_space)
next
 assume R: ?rhs
 show ?lhs
 proof (cases "\i\I. X i = trivial_topology")
 case False
 show ?thesis
 unfolding completely_regular_space_alt'
 proof clarify
 fix W x
 assume W: "openin (product_topology X I) W" and "x \ W"
 then obtain U where finU: "finite {i \ I. U i \ topspace (X i)}"
 and ope: "\i. i\I \ openin (X i) (U i)"
 and x: "x \ Pi\<^sub>E I U" and "Pi\<^sub>E I U \ W"
 by (auto simp: openin_product_topology_alt)
 have "\i \ I. openin (X i) (U i) \ x i \ U i
 \<longrightarrow> (\<exists>f. continuous_map (X i) euclideanreal f \<and>
 f (x i) = 0 \<and> (\<forall>x \<in> topspace(X i). x \<notin> U i \<longrightarrow> f x = 1))"
 using R unfolding completely_regular_space_alt'
 by (smt (verit) DiffI False image_subset_iff singletonD)
 with ope x have "\i. \f. i \ I \ continuous_map (X i) euclideanreal f \
 f (x i) = 0 \<and> (\<forall>x \<in> topspace (X i). x \<notin> U i \<longrightarrow> f x = 1)"
 by auto
 then obtain f where f: "\i. i \ I \ continuous_map (X i) euclideanreal (f i) \
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