(*  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/Countable_Set"

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

  Linear_Algebra

  Norm_Arith

begin

lemma dist_0_norm:

  fixes x :: "'a::real_normed_vector"

  shows "dist 0 x = norm x"

unfolding dist_norm by simp

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)

(* LEGACY *)

lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s \<circ> r) ----> l"

  by (rule LIMSEQ_subseq_LIMSEQ)

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)

lemma Lim_within_open:

  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"

  shows "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"

  by (fact tendsto_within_open)

lemma continuous_on_union:

  "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"

  by (fact continuous_on_closed_Un)

lemma continuous_on_cases:

  "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t g \<Longrightarrow>

    \<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x \<Longrightarrow>

    continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"

  by (rule continuous_on_If) auto

subsection {* Topological Basis *}

context topological_space

begin

definition "topological_basis B \<longleftrightarrow>

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

lemma topological_basis:

  "topological_basis B \<longleftrightarrow> (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"

  unfolding topological_basis_def

  apply safe

     apply fastforce

    apply fastforce

   apply (erule_tac x="x" in allE)

   apply simp

   apply (rule_tac x="{x}" in exI)

  apply auto

  done

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'"

  then have "(\<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

  then show "\<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)

    then show "\<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'"

    and "\<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"

    and "open O'"

    and "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"

    and "X \<in> B"

  shows "open X"

  using assms by (simp add: topological_basis_def)

lemma topological_basis_imp_subbasis:

  assumes B: "topological_basis B"

  shows "open = generate_topology B"

proof (intro ext iffI)

  fix S :: "'a set"

  assume "open S"

  with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"

    unfolding topological_basis_def by blast

  then show "generate_topology B S"

    by (auto intro: generate_topology.intros dest: topological_basis_open)

next

  fix S :: "'a set"

  assume "generate_topology B S"

  then show "open S"

    by induct (auto dest: topological_basis_open[OF B])

qed

lemma basis_dense:

  fixes B :: "'a set set"

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

  assumes "topological_basis B"

    and 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" and "X \<noteq> {}"

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

  obtain B' where "B' \<in> B" "f X \<in> B'" "B' \<subseteq> X" .

  then show "\<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 A a obtain A0 where "A0 \<in> A" "x \<in> A0" "A0 \<subseteq> a"

      by (rule topological_basisE)

    moreover

    from B b obtain B0 where "B0 \<in> B" "y \<in> B0" "B0 \<subseteq> b"

      by (rule topological_basisE)

    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"

    and 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:

  "\<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

end

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]

  apply atomize_elim

  apply (elim exE)

  apply (rule_tac x="range A" in exI)

  apply auto

  done

lemma (in first_countable_topology) first_countable_basis_Int_stableE:

  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)"

    "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A"

proof atomize_elim

  obtain A' where A':

    "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"

    by (rule first_countable_basisE) blast

  def A \<equiv> "(\<lambda>N. \<Inter>((\<lambda>n. from_nat_into A' n) ` N)) ` (Collect finite::nat set set)"

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

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

  proof (safe intro!: exI[where x=A])

    show "countable A"

      unfolding A_def by (intro countable_image countable_Collect_finite)

    fix a

    assume "a \<in> A"

    then show "x \<in> a" "open a"

      using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)

  next

    let ?int = "\<lambda>N. \<Inter>(from_nat_into A' ` N)"

    fix a b

    assume "a \<in> A" "b \<in> A"

    then obtain N M where "a = ?int N" "b = ?int M" "finite (N \<union> M)"

      by (auto simp: A_def)

    then show "a \<inter> b \<in> A"

      by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"])

  next

    fix S

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

    then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast

    then show "\<exists>a\<in>A. a \<subseteq> S" using a A'

      by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"])

  qed

qed

lemma (in topological_space) first_countableI:

  assumes "countable A"

    and 1: "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"

    and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"

  shows "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"

proof (safe intro!: exI[of _ "from_nat_into A"])

  fix i

  have "A \<noteq> {}" using 2[of UNIV] by auto

  show "x \<in> from_nat_into A i" "open (from_nat_into A i)"

    using range_from_nat_into_subset[OF `A \<noteq> {}`] 1 by auto

next

  fix S

  assume "open S" "x\<in>S" from 2[OF this]

  show "\<exists>i. from_nat_into A i \<subseteq> S"

    using subset_range_from_nat_into[OF `countable A`] by auto

qed

instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology

proof

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

  obtain A where A:

      "countable A"

      "\<And>a. a \<in> A \<Longrightarrow> fst x \<in> a"

      "\<And>a. a \<in> A \<Longrightarrow> open a"

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

    by (rule first_countable_basisE[of "fst x"]) blast

  obtain B where B:

      "countable B"

      "\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a"

      "\<And>a. a \<in> B \<Longrightarrow> open a"

      "\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S"

    by (rule first_countable_basisE[of "snd x"]) blast

  show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set.

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

  proof (rule first_countableI[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" and "open (a \<times> b)"

      unfolding mem_Times_iff

      by (auto intro: open_Times)

  next

    fix S

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

    then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S"

      by (rule open_prod_elim)

    moreover

    from a'b' 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

class second_countable_topology = topological_space +

  assumes ex_countable_subbasis:

    "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"

begin

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

proof -

  from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"

    by blast

  let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"

  show ?thesis

  proof (intro exI conjI)

    show "countable ?B"

      by (intro countable_image countable_Collect_finite_subset B)

    {

      fix S

      assume "open S"

      then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"

        unfolding B

      proof induct

        case UNIV

        show ?case by (intro exI[of _ "{{}}"]) simp

      next

        case (Int a b)

        then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"

          and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"

          by blast

        show ?case

          unfolding x y Int_UN_distrib2

          by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))

      next

        case (UN K)

        then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto

        then obtain k where

            "\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> UNION (k ka) Inter = ka"

          unfolding bchoice_iff ..

        then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K"

          by (intro exI[of _ "UNION K k"]) auto

      next

        case (Basis S)

        then show ?case

          by (intro exI[of _ "{{S}}"]) auto

      qed

      then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"

        unfolding subset_image_iff by blast }

    then show "topological_basis ?B"

      unfolding topological_space_class.topological_basis_def

      by (safe intro!: topological_space_class.open_Inter)

         (simp_all add: B generate_topology.Basis subset_eq)

  qed

qed

end

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> open = generate_topology B"

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

      topological_basis_imp_subbasis)

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::nat \<Rightarrow> 'a set.

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

    by (intro first_countableI[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"

  then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp

next

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

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

  then have "topology (openin T1) = topology (openin T2)" by simp

  then show "T1 = T2" unfolding openin_inverse .

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)"

      by blast

    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

  by (auto simp add: openin_subtopology)

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

  unfolding openin_subtopology

  by auto (metis IntD1 in_mono openin_subset)

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"

      then have "\<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)

  apply (auto simp add: open_openin[symmetric])

  done

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 \<Longrightarrow> 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"

  by (metis closedin_closed inf.absorb2)

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

  then show ?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)

    by (metis dist_commute dist_triangle_lt)

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

    unfolding T_def 

    by auto (metis dist_self)

  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 \<Longrightarrow>

    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 \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"

  by (simp add: subset_eq)

lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> 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)"

proof -

  have "open (dist x -` {..<e})"

    by (intro open_vimage open_lessThan continuous_on_intros)

  also have "dist x -` {..<e} = ball x e"

    by auto

  finally show ?thesis .

qed

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)

  apply (metis zero_le_dist order_trans dist_self)

  done

lemma ball_empty[intro]: "e \<le> 0 \<Longrightarrow> 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 (in euclidean_space) eucl_less (infix "<e" 50)

  where "eucl_less a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"

definition box_eucl_less: "box a b = {x. a <e x \<and> x <e b}"

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

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

  and in_box_eucl_less: "x \<in> box a b \<longleftrightarrow> a <e x \<and> x <e b"

  and mem_box: "x \<in> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i)"

    "x \<in> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"

  by (auto simp: box_eucl_less eucl_less_def cbox_def)

lemma mem_box_real[simp]:

  "(x::real) \<in> box a b \<longleftrightarrow> a < x \<and> x < b"

  "(x::real) \<in> cbox a b \<longleftrightarrow> a \<le> x \<and> x \<le> b"

  by (auto simp: mem_box)

lemma box_real[simp]:

  fixes a b:: real

  shows "box a b = {a <..< b}" "cbox a b = {a .. b}"

  by auto

lemma rational_boxes:

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

  assumes "e > 0"

  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: "e' > 0"

    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] obtain a where

    a: "\<forall>xa. a xa \<in> \<rat> \<and> a xa < x \<bullet> xa \<and> x \<bullet> xa - a xa < e'" ..

  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] obtain b where

    b: "\<forall>xa. b xa \<in> \<rat> \<and> x \<bullet> xa < b xa \<and> b xa - x \<bullet> xa < e'" ..

  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))\<^sup>2)"

      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))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"

        by (rule power_strict_mono) auto

      then show "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < e\<^sup>2 / 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>\<^sub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"

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

proof -

  {

    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 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 have "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)

  }

  then show ?thesis by (auto simp: I_def)

qed

subsection{* Connectedness *}

lemma connected_local:

 "connected S \<longleftrightarrow>

  \<not> (\<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 \<noteq> {} \<and>

      e2 \<noteq> {})"

  unfolding connected_def openin_open

  by 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"

    then have ?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

    by (metis double_complement)

  then have th0: "connected S \<longleftrightarrow>

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

    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")

    apply (simp add: closed_def)

    apply metis

    done

  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