(*  title:      HOL/Library/Topology_Euclidian_Space.thy

    Author:     Amine Chaieb, University of Cambridge

    Author:     Robert Himmelmann, TU Muenchen

    Author:     Brian Huffman, Portland State University

*)

header {* Elementary topology in Euclidean space. *}

theory Topology_Euclidean_Space

imports

  Complex_Main

  "~~/src/HOL/Library/Diagonal_Subsequence"

  "~~/src/HOL/Library/Countable_Set"

  "~~/src/HOL/Library/Glbs"

  "~~/src/HOL/Library/FuncSet"

  Linear_Algebra

  Norm_Arith

begin

lemma dist_double: "dist x y < d / 2 \<Longrightarrow> dist x z < d / 2 \<Longrightarrow> dist y z < d"

  using dist_triangle[of y z x] by (simp add: dist_commute)

(* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)

lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"

  apply (frule isGlb_isLb)

  apply (frule_tac x = y in isGlb_isLb)

  apply (blast intro!: order_antisym dest!: isGlb_le_isLb)

  done

lemma countable_PiE: 

  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (PiE I F)"

  by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)

subsection {* Topological Basis *}

context topological_space

begin

definition "topological_basis B =

  ((\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> Union B' = x)))"

lemma topological_basis_iff:

  assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"

  shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"

    (is "_ \<longleftrightarrow> ?rhs")

proof safe

  fix O' and x::'a

  assume H: "topological_basis B" "open O'" "x \<in> O'"

  hence "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)

  then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto

  thus "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto

next

  assume H: ?rhs

  show "topological_basis B" using assms unfolding topological_basis_def

  proof safe

    fix O'::"'a set" assume "open O'"

    with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"

      by (force intro: bchoice simp: Bex_def)

    thus "\<exists>B'\<subseteq>B. \<Union>B' = O'"

      by (auto intro: exI[where x="{f x |x. x \<in> O'}"])

  qed

qed

lemma topological_basisI:

  assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"

  assumes "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"

  shows "topological_basis B"

  using assms by (subst topological_basis_iff) auto

lemma topological_basisE:

  fixes O'

  assumes "topological_basis B"

  assumes "open O'"

  assumes "x \<in> O'"

  obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"

proof atomize_elim

  from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'" by (simp add: topological_basis_def)

  with topological_basis_iff assms

  show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'" using assms by (simp add: Bex_def)

qed

lemma topological_basis_open:

  assumes "topological_basis B"

  assumes "X \<in> B"

  shows "open X"

  using assms

  by (simp add: topological_basis_def)

lemma basis_dense:

  fixes B::"'a set set" and f::"'a set \<Rightarrow> 'a"

  assumes "topological_basis B"

  assumes choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"

  shows "(\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X))"

proof (intro allI impI)

  fix X::"'a set" assume "open X" "X \<noteq> {}"

  from topological_basisE[OF `topological_basis B` `open X` choosefrom_basis[OF `X \<noteq> {}`]]

  guess B' . note B' = this

  thus "\<exists>B'\<in>B. f B' \<in> X" by (auto intro!: choosefrom_basis)

qed

end

lemma topological_basis_prod:

  assumes A: "topological_basis A" and B: "topological_basis B"

  shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"

  unfolding topological_basis_def

proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])

  fix S :: "('a \<times> 'b) set" assume "open S"

  then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"

  proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])

    fix x y assume "(x, y) \<in> S"

    from open_prod_elim[OF `open S` this]

    obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"

      by (metis mem_Sigma_iff)

    moreover from topological_basisE[OF A a] guess A0 .

    moreover from topological_basisE[OF B b] guess B0 .

    ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"

      by (intro UN_I[of "(A0, B0)"]) auto

  qed auto

qed (metis A B topological_basis_open open_Times)

subsection {* Countable Basis *}

locale countable_basis =

  fixes B::"'a::topological_space set set"

  assumes is_basis: "topological_basis B"

  assumes countable_basis: "countable B"

begin

lemma open_countable_basis_ex:

  assumes "open X"

  shows "\<exists>B' \<subseteq> B. X = Union B'"

  using assms countable_basis is_basis unfolding topological_basis_def by blast

lemma open_countable_basisE:

  assumes "open X"

  obtains B' where "B' \<subseteq> B" "X = Union B'"

  using assms open_countable_basis_ex by (atomize_elim) simp

lemma countable_dense_exists:

  shows "\<exists>D::'a set. countable D \<and> (\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"

proof -

  let ?f = "(\<lambda>B'. SOME x. x \<in> B')"

  have "countable (?f ` B)" using countable_basis by simp

  with basis_dense[OF is_basis, of ?f] show ?thesis

    by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)

qed

lemma countable_dense_setE:

  obtains D :: "'a set"

  where "countable D" "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"

  using countable_dense_exists by blast

text {* Construction of an increasing sequence approximating open sets,

  therefore basis which is closed under union. *}

definition union_closed_basis::"'a set set" where

  "union_closed_basis = (\<lambda>l. \<Union>set l) ` lists B"

lemma basis_union_closed_basis: "topological_basis union_closed_basis"

proof (rule topological_basisI)

  fix O' and x::'a assume "open O'" "x \<in> O'"

  from topological_basisE[OF is_basis this] guess B' . note B' = this

  thus "\<exists>B'\<in>union_closed_basis. x \<in> B' \<and> B' \<subseteq> O'" unfolding union_closed_basis_def

    by (auto intro!: bexI[where x="[B']"])

next

  fix B' assume "B' \<in> union_closed_basis"

  thus "open B'"

    using topological_basis_open[OF is_basis]

    by (auto simp: union_closed_basis_def)

qed

lemma countable_union_closed_basis: "countable union_closed_basis"

  unfolding union_closed_basis_def using countable_basis by simp

lemmas open_union_closed_basis = topological_basis_open[OF basis_union_closed_basis]

lemma union_closed_basis_ex:

 assumes X: "X \<in> union_closed_basis"

 shows "\<exists>B'. finite B' \<and> X = \<Union>B' \<and> B' \<subseteq> B"

proof -

  from X obtain l where "\<And>x. x\<in>set l \<Longrightarrow> x\<in>B" "X = \<Union>set l" by (auto simp: union_closed_basis_def)

  thus ?thesis by auto

qed

lemma union_closed_basisE:

  assumes "X \<in> union_closed_basis"

  obtains B' where "finite B'" "X = \<Union>B'" "B' \<subseteq> B" using union_closed_basis_ex[OF assms] by blast

lemma union_closed_basisI:

  assumes "finite B'" "X = \<Union>B'" "B' \<subseteq> B"

  shows "X \<in> union_closed_basis"

proof -

  from finite_list[OF `finite B'`] guess l ..

  thus ?thesis using assms unfolding union_closed_basis_def by (auto intro!: image_eqI[where x=l])

qed

lemma empty_basisI[intro]: "{} \<in> union_closed_basis"

  by (rule union_closed_basisI[of "{}"]) auto

lemma union_basisI[intro]:

  assumes "X \<in> union_closed_basis" "Y \<in> union_closed_basis"

  shows "X \<union> Y \<in> union_closed_basis"

  using assms by (auto intro: union_closed_basisI elim!:union_closed_basisE)

lemma open_imp_Union_of_incseq:

  assumes "open X"

  shows "\<exists>S. incseq S \<and> (\<Union>j. S j) = X \<and> range S \<subseteq> union_closed_basis"

proof -

  from open_countable_basis_ex[OF `open X`]

  obtain B' where B': "B'\<subseteq>B" "X = \<Union>B'" by auto

  from this(1) countable_basis have "countable B'" by (rule countable_subset)

  show ?thesis

  proof cases

    assume "B' \<noteq> {}"

    def S \<equiv> "\<lambda>n. \<Union>i\<in>{0..n}. from_nat_into B' i"

    have S:"\<And>n. S n = \<Union>{from_nat_into B' i|i. i\<in>{0..n}}" unfolding S_def by force

    have "incseq S" by (force simp: S_def incseq_Suc_iff)

    moreover

    have "(\<Union>j. S j) = X" unfolding B'

    proof safe

      fix x X assume "X \<in> B'" "x \<in> X"

      then obtain n where "X = from_nat_into B' n"

        by (metis `countable B'` from_nat_into_surj)

      also have "\<dots> \<subseteq> S n" by (auto simp: S_def)

      finally show "x \<in> (\<Union>j. S j)" using `x \<in> X` by auto

    next

      fix x n

      assume "x \<in> S n"

      also have "\<dots> = (\<Union>i\<in>{0..n}. from_nat_into B' i)"

        by (simp add: S_def)

      also have "\<dots> \<subseteq> (\<Union>i. from_nat_into B' i)" by auto

      also have "\<dots> \<subseteq> \<Union>B'" using `B' \<noteq> {}` by (auto intro: from_nat_into)

      finally show "x \<in> \<Union>B'" .

    qed

    moreover have "range S \<subseteq> union_closed_basis" using B'

      by (auto intro!: union_closed_basisI[OF _ S] simp: from_nat_into `B' \<noteq> {}`)

    ultimately show ?thesis by auto

  qed (auto simp: B')

qed

lemma open_incseqE:

  assumes "open X"

  obtains S where "incseq S" "(\<Union>j. S j) = X" "range S \<subseteq> union_closed_basis"

  using open_imp_Union_of_incseq assms by atomize_elim

end

class first_countable_topology = topological_space +

  assumes first_countable_basis:

    "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"

lemma (in first_countable_topology) countable_basis_at_decseq:

  obtains A :: "nat \<Rightarrow> 'a set" where

    "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"

    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

proof atomize_elim

  from first_countable_basis[of x] obtain A

    where "countable A"

    and nhds: "\<And>a. a \<in> A \<Longrightarrow> open a" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a"

    and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"  by auto

  then have "A \<noteq> {}" by auto

  with `countable A` have r: "A = range (from_nat_into A)" by auto

  def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. from_nat_into A i"

  show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>

      (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"

  proof (safe intro!: exI[of _ F])

    fix i

    show "open (F i)" using nhds(1) r by (auto simp: F_def intro!: open_INT)

    show "x \<in> F i" using nhds(2) r by (auto simp: F_def)

  next

    fix S assume "open S" "x \<in> S"

    from incl[OF this] obtain i where "F i \<subseteq> S"

      by (subst (asm) r) (auto simp: F_def)

    moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"

      by (auto simp: F_def)

    ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"

      by (auto simp: eventually_sequentially)

  qed

qed

lemma (in first_countable_topology) first_countable_basisE:

  obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"

    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"

  using first_countable_basis[of x]

  by atomize_elim auto

instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology

proof

  fix x :: "'a \<times> 'b"

  from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this

  from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this

  show "\<exists>A::('a\<times>'b) set set. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"

  proof (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)

    fix a b assume x: "a \<in> A" "b \<in> B"

    with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" "open (a \<times> b)"

      unfolding mem_Times_iff by (auto intro: open_Times)

  next

    fix S assume "open S" "x \<in> S"

    from open_prod_elim[OF this] guess a' b' .

    moreover with A(4)[of a'] B(4)[of b']

    obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'" by auto

    ultimately show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (A \<times> B). a \<subseteq> S"

      by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])

  qed (simp add: A B)

qed

instance metric_space \<subseteq> first_countable_topology

proof

  fix x :: 'a

  show "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"

  proof (intro exI, safe)

    fix S assume "open S" "x \<in> S"

    then obtain r where "0 < r" "{y. dist x y < r} \<subseteq> S"

      by (auto simp: open_dist dist_commute subset_eq)

    moreover from reals_Archimedean[OF `0 < r`] guess n ..

    moreover

    then have "{y. dist x y < inverse (Suc n)} \<subseteq> {y. dist x y < r}"

      by (auto simp: inverse_eq_divide)

    ultimately show "\<exists>a\<in>range (\<lambda>n. {y. dist x y < inverse (Suc n)}). a \<subseteq> S"

      by auto

  qed (auto intro: open_ball)

qed

class second_countable_topology = topological_space +

  assumes ex_countable_basis:

    "\<exists>B::'a::topological_space set set. countable B \<and> topological_basis B"

sublocale second_countable_topology < countable_basis "SOME B. countable B \<and> topological_basis B"

  using someI_ex[OF ex_countable_basis] by unfold_locales safe

instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology

proof

  obtain A :: "'a set set" where "countable A" "topological_basis A"

    using ex_countable_basis by auto

  moreover

  obtain B :: "'b set set" where "countable B" "topological_basis B"

    using ex_countable_basis by auto

  ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> topological_basis B"

    by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod)

qed

instance second_countable_topology \<subseteq> first_countable_topology

proof

  fix x :: 'a

  def B \<equiv> "SOME B::'a set set. countable B \<and> topological_basis B"

  then have B: "countable B" "topological_basis B"

    using countable_basis is_basis

    by (auto simp: countable_basis is_basis)

  then show "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"

    by (intro exI[of _ "{b\<in>B. x \<in> b}"])

       (fastforce simp: topological_space_class.topological_basis_def)

qed

subsection {* Polish spaces *}

text {* Textbooks define Polish spaces as completely metrizable.

  We assume the topology to be complete for a given metric. *}

class polish_space = complete_space + second_countable_topology

subsection {* General notion of a topology as a value *}

definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"

typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"

  morphisms "openin" "topology"

  unfolding istopology_def by blast

lemma istopology_open_in[intro]: "istopology(openin U)"

  using openin[of U] by blast

lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"

  using topology_inverse[unfolded mem_Collect_eq] .

lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"

  using topology_inverse[of U] istopology_open_in[of "topology U"] by auto

lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"

proof-

  { assume "T1=T2"

    hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp }

  moreover

  { assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"

    hence "openin T1 = openin T2" by (simp add: fun_eq_iff)

    hence "topology (openin T1) = topology (openin T2)" by simp

    hence "T1 = T2" unfolding openin_inverse .

  }

  ultimately show ?thesis by blast

qed

text{* Infer the "universe" from union of all sets in the topology. *}

definition "topspace T =  \<Union>{S. openin T S}"

subsubsection {* Main properties of open sets *}

lemma openin_clauses:

  fixes U :: "'a topology"

  shows "openin U {}"

  "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"

  "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"

  using openin[of U] unfolding istopology_def mem_Collect_eq

  by fast+

lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"

  unfolding topspace_def by blast

lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)

lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"

  using openin_clauses by simp

lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"

  using openin_clauses by simp

lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"

  using openin_Union[of "{S,T}" U] by auto

lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)

lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"

  (is "?lhs \<longleftrightarrow> ?rhs")

proof

  assume ?lhs

  then show ?rhs by auto

next

  assume H: ?rhs

  let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"

  have "openin U ?t" by (simp add: openin_Union)

  also have "?t = S" using H by auto

  finally show "openin U S" .

qed

subsubsection {* Closed sets *}

definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"

lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)

lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)

lemma closedin_topspace[intro,simp]:

  "closedin U (topspace U)" by (simp add: closedin_def)

lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"

  by (auto simp add: Diff_Un closedin_def)

lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto

lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"

  shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto

lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"

  using closedin_Inter[of "{S,T}" U] by auto

lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast

lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"

  apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

  apply (metis openin_subset subset_eq)

  done

lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"

  by (simp add: openin_closedin_eq)

lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"

proof-

  have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT

    by (auto simp add: topspace_def openin_subset)

  then show ?thesis using oS cT by (auto simp add: closedin_def)

qed

lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"

proof-

  have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT

    by (auto simp add: topspace_def )

  then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)

qed

subsubsection {* Subspace topology *}

definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"

lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"

  (is "istopology ?L")

proof-

  have "?L {}" by blast

  {fix A B assume A: "?L A" and B: "?L B"

    from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V" by blast

    have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+

    then have "?L (A \<inter> B)" by blast}

  moreover

  {fix K assume K: "K \<subseteq> Collect ?L"

    have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"

      apply (rule set_eqI)

      apply (simp add: Ball_def image_iff)

      by metis

    from K[unfolded th0 subset_image_iff]

    obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast

    have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto

    moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq)

    ultimately have "?L (\<Union>K)" by blast}

  ultimately show ?thesis

    unfolding subset_eq mem_Collect_eq istopology_def by blast

qed

lemma openin_subtopology:

  "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"

  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

  by auto

lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"

  by (auto simp add: topspace_def openin_subtopology)

lemma closedin_subtopology:

  "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"

  unfolding closedin_def topspace_subtopology

  apply (simp add: openin_subtopology)

  apply (rule iffI)

  apply clarify

  apply (rule_tac x="topspace U - T" in exI)

  by auto

lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"

  unfolding openin_subtopology

  apply (rule iffI, clarify)

  apply (frule openin_subset[of U])  apply blast

  apply (rule exI[where x="topspace U"])

  apply auto

  done

lemma subtopology_superset:

  assumes UV: "topspace U \<subseteq> V"

  shows "subtopology U V = U"

proof-

  {fix S

    {fix T assume T: "openin U T" "S = T \<inter> V"

      from T openin_subset[OF T(1)] UV have eq: "S = T" by blast

      have "openin U S" unfolding eq using T by blast}

    moreover

    {assume S: "openin U S"

      hence "\<exists>T. openin U T \<and> S = T \<inter> V"

        using openin_subset[OF S] UV by auto}

    ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}

  then show ?thesis unfolding topology_eq openin_subtopology by blast

qed

lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"

  by (simp add: subtopology_superset)

lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"

  by (simp add: subtopology_superset)

subsubsection {* The standard Euclidean topology *}

definition

  euclidean :: "'a::topological_space topology" where

  "euclidean = topology open"

lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"

  unfolding euclidean_def

  apply (rule cong[where x=S and y=S])

  apply (rule topology_inverse[symmetric])

  apply (auto simp add: istopology_def)

  done

lemma topspace_euclidean: "topspace euclidean = UNIV"

  apply (simp add: topspace_def)

  apply (rule set_eqI)

  by (auto simp add: open_openin[symmetric])

lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"

  by (simp add: topspace_euclidean topspace_subtopology)

lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"

  by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)

lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"

  by (simp add: open_openin openin_subopen[symmetric])

text {* Basic "localization" results are handy for connectedness. *}

lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"

  by (auto simp add: openin_subtopology open_openin[symmetric])

lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"

  by (auto simp add: openin_open)

lemma open_openin_trans[trans]:

 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"

  by (metis Int_absorb1  openin_open_Int)

lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"

  by (auto simp add: openin_open)

lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"

  by (simp add: closedin_subtopology closed_closedin Int_ac)

lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"

  by (metis closedin_closed)

lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"

  apply (subgoal_tac "S \<inter> T = T" )

  apply auto

  apply (frule closedin_closed_Int[of T S])

  by simp

lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"

  by (auto simp add: closedin_closed)

lemma openin_euclidean_subtopology_iff:

  fixes S U :: "'a::metric_space set"

  shows "openin (subtopology euclidean U) S

  \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")

proof

  assume ?lhs thus ?rhs unfolding openin_open open_dist by blast

next

  def T \<equiv> "{x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"

  have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"

    unfolding T_def

    apply clarsimp

    apply (rule_tac x="d - dist x a" in exI)

    apply (clarsimp simp add: less_diff_eq)

    apply (erule rev_bexI)

    apply (rule_tac x=d in exI, clarify)

    apply (erule le_less_trans [OF dist_triangle])

    done

  assume ?rhs hence 2: "S = U \<inter> T"

    unfolding T_def

    apply auto

    apply (drule (1) bspec, erule rev_bexI)

    apply auto

    done

  from 1 2 show ?lhs

    unfolding openin_open open_dist by fast

qed

text {* These "transitivity" results are handy too *}

lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T

  \<Longrightarrow> openin (subtopology euclidean U) S"

  unfolding open_openin openin_open by blast

lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"

  by (auto simp add: openin_open intro: openin_trans)

lemma closedin_trans[trans]:

 "closedin (subtopology euclidean T) S \<Longrightarrow>

           closedin (subtopology euclidean U) T

           ==> closedin (subtopology euclidean U) S"

  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)

lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"

  by (auto simp add: closedin_closed intro: closedin_trans)

subsection {* Open and closed balls *}

definition

  ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where

  "ball x e = {y. dist x y < e}"

definition

  cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where

  "cball x e = {y. dist x y \<le> e}"

lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"

  by (simp add: ball_def)

lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"

  by (simp add: cball_def)

lemma mem_ball_0:

  fixes x :: "'a::real_normed_vector"

  shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"

  by (simp add: dist_norm)

lemma mem_cball_0:

  fixes x :: "'a::real_normed_vector"

  shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"

  by (simp add: dist_norm)

lemma centre_in_ball: "x \<in> ball x e \<longleftrightarrow> 0 < e"

  by simp

lemma centre_in_cball: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"

  by simp

lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)

lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)

lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)

lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"

  by (simp add: set_eq_iff) arith

lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"

  by (simp add: set_eq_iff)

lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"

  "(a::real) - b < 0 \<longleftrightarrow> a < b"

  "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+

lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"

  "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+

lemma open_ball[intro, simp]: "open (ball x e)"

  unfolding open_dist ball_def mem_Collect_eq Ball_def

  unfolding dist_commute

  apply clarify

  apply (rule_tac x="e - dist xa x" in exI)

  using dist_triangle_alt[where z=x]

  apply (clarsimp simp add: diff_less_iff)

  apply atomize

  apply (erule_tac x="y" in allE)

  apply (erule_tac x="xa" in allE)

  by arith

lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"

  unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

lemma openE[elim?]:

  assumes "open S" "x\<in>S" 

  obtains e where "e>0" "ball x e \<subseteq> S"

  using assms unfolding open_contains_ball by auto

lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

  by (metis open_contains_ball subset_eq centre_in_ball)

lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"

  unfolding mem_ball set_eq_iff

  apply (simp add: not_less)

  by (metis zero_le_dist order_trans dist_self)

lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp

lemma euclidean_dist_l2:

  fixes x y :: "'a :: euclidean_space"

  shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"

  unfolding dist_norm norm_eq_sqrt_inner setL2_def

  by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)

definition "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"

lemma rational_boxes:

  fixes x :: "'a\<Colon>euclidean_space"

  assumes "0 < e"

  shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat> ) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e"

proof -

  def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"

  then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos simp: DIM_positive)

  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")

  proof

    fix i from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e show "?th i" by auto

  qed

  from choice[OF this] guess a .. note a = this

  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")

  proof

    fix i from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e show "?th i" by auto

  qed

  from choice[OF this] guess b .. note b = this

  let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"

  show ?thesis

  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)

    fix y :: 'a assume *: "y \<in> box ?a ?b"

    have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<twosuperior>)"

      unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

    also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"

    proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

      fix i :: "'a" assume i: "i \<in> Basis"

      have "a i < y\<bullet>i \<and> y\<bullet>i < b i" using * i by (auto simp: box_def)

      moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'" using a by auto

      moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'" using b by auto

      ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'" by auto

      then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"

        unfolding e'_def by (auto simp: dist_real_def)

      then have "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"

        by (rule power_strict_mono) auto

      then show "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < e\<twosuperior> / real DIM('a)"

        by (simp add: power_divide)

    qed auto

    also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat)

    finally show "y \<in> ball x e" by (auto simp: ball_def)

  qed (insert a b, auto simp: box_def)

qed

lemma open_UNION_box:

  fixes M :: "'a\<Colon>euclidean_space set"

  assumes "open M" 

  defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"

  defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"

  defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^isub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"

  shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"

proof safe

  fix x assume "x \<in> M"

  obtain e where e: "e > 0" "ball x e \<subseteq> M"

    using openE[OF `open M` `x \<in> M`] by auto

  moreover then obtain a b where ab: "x \<in> box a b"

    "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>" "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>" "box a b \<subseteq> ball x e"

    using rational_boxes[OF e(1)] by metis

  ultimately show "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))"

     by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])

        (auto simp: euclidean_representation I_def a'_def b'_def)

qed (auto simp: I_def)

subsection{* Connectedness *}

definition "connected S \<longleftrightarrow>

  ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})

  \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"

lemma connected_local:

 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.

                 openin (subtopology euclidean S) e1 \<and>

                 openin (subtopology euclidean S) e2 \<and>

                 S \<subseteq> e1 \<union> e2 \<and>

                 e1 \<inter> e2 = {} \<and>

                 ~(e1 = {}) \<and>

                 ~(e2 = {}))"

unfolding connected_def openin_open by (safe, blast+)

lemma exists_diff:

  fixes P :: "'a set \<Rightarrow> bool"

  shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")

proof-

  {assume "?lhs" hence ?rhs by blast }

  moreover

  {fix S assume H: "P S"

    have "S = - (- S)" by auto

    with H have "P (- (- S))" by metis }

  ultimately show ?thesis by metis

qed

lemma connected_clopen: "connected S \<longleftrightarrow>

        (\<forall>T. openin (subtopology euclidean S) T \<and>

            closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")

proof-

  have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (- e2) \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"

    unfolding connected_def openin_open closedin_closed

    apply (subst exists_diff) by blast

  hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"

    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis

  have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"

    (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")

    unfolding connected_def openin_open closedin_closed by auto

  {fix e2

    {fix e1 have "?P e2 e1 \<longleftrightarrow> (\<exists>t.  closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"

        by auto}

    then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}

  then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast

  then show ?thesis unfolding th0 th1 by simp

qed

lemma connected_empty[simp, intro]: "connected {}"

  by (simp add: connected_def)

subsection{* Limit points *}

definition (in topological_space)

  islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where

  "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"

lemma islimptI:

  assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  shows "x islimpt S"

  using assms unfolding islimpt_def by auto

lemma islimptE:

  assumes "x islimpt S" and "x \<in> T" and "open T"

  obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"

  using assms unfolding islimpt_def by auto

lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"

  unfolding islimpt_def eventually_at_topological by auto

lemma islimpt_subset: "\<lbrakk>x islimpt S; S \<subseteq> T\<rbrakk> \<Longrightarrow> x islimpt T"

  unfolding islimpt_def by fast

lemma islimpt_approachable:

  fixes x :: "'a::metric_space"

  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"

  unfolding islimpt_iff_eventually eventually_at by fast

lemma islimpt_approachable_le:

  fixes x :: "'a::metric_space"

  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"

  unfolding islimpt_approachable

  using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",

    THEN arg_cong [where f=Not]]

  by (simp add: Bex_def conj_commute conj_left_commute)

lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"

  unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)

text {* A perfect space has no isolated points. *}

lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"

  unfolding islimpt_UNIV_iff by (rule not_open_singleton)

lemma perfect_choose_dist:

  fixes x :: "'a::{perfect_space, metric_space}"

  shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"

using islimpt_UNIV [of x]

by (simp add: islimpt_approachable)

lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"

  unfolding closed_def

  apply (subst open_subopen)

  apply (simp add: islimpt_def subset_eq)

  by (metis ComplE ComplI)

lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"

  unfolding islimpt_def by auto

lemma finite_set_avoid:

  fixes a :: "'a::metric_space"

  assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"

proof(induct rule: finite_induct[OF fS])

  case 1 thus ?case by (auto intro: zero_less_one)

next

  case (2 x F)

  from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast

  {assume "x = a" hence ?case using d by auto  }

  moreover

  {assume xa: "x\<noteq>a"

    let ?d = "min d (dist a x)"

    have dp: "?d > 0" using xa d(1) using dist_nz by auto

    from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto

    with dp xa have ?case by(auto intro!: exI[where x="?d"]) }

  ultimately show ?case by blast

qed

lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"

  by (simp add: islimpt_iff_eventually eventually_conj_iff)

lemma discrete_imp_closed:

  fixes S :: "'a::metric_space set"

  assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"

  shows "closed S"

proof-

  {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"

    from e have e2: "e/2 > 0" by arith

    from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast

    let ?m = "min (e/2) (dist x y) "

    from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])

    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast

    have th: "dist z y < e" using z y

      by (intro dist_triangle_lt [where z=x], simp)

    from d[rule_format, OF y(1) z(1) th] y z

    have False by (auto simp add: dist_commute)}

  then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])

qed

subsection {* Interior of a Set *}

definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"

lemma interiorI [intro?]:

  assumes "open T" and "x \<in> T" and "T \<subseteq> S"

  shows "x \<in> interior S"

  using assms unfolding interior_def by fast

lemma interiorE [elim?]:

  assumes "x \<in> interior S"

  obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"

  using assms unfolding interior_def by fast

lemma open_interior [simp, intro]: "open (interior S)"

  by (simp add: interior_def open_Union)

lemma interior_subset: "interior S \<subseteq> S"

  by (auto simp add: interior_def)

lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"

  by (auto simp add: interior_def)

lemma interior_open: "open S \<Longrightarrow> interior S = S"

  by (intro equalityI interior_subset interior_maximal subset_refl)

lemma interior_eq: "interior S = S \<longleftrightarrow> open S"

  by (metis open_interior interior_open)

lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"

  by (metis interior_maximal interior_subset subset_trans)

lemma interior_empty [simp]: "interior {} = {}"

  using open_empty by (rule interior_open)

lemma interior_UNIV [simp]: "interior UNIV = UNIV"

  using open_UNIV by (rule interior_open)

lemma interior_interior [simp]: "interior (interior S) = interior S"

  using open_interior by (rule interior_open)

lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"

  by (auto simp add: interior_def)

lemma interior_unique: