Theory Topology_ZF_examples

theory Topology_ZF_examples
imports Topology_ZF Cardinal_ZF
(* 
    This file is a part of IsarMathLib - 
    a library of formalized mathematics written for Isabelle/Isar.

    Copyright (C) 2012 Daniel de la Concepcion

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header{*\isaheader{Topology\_ZF\_examples.thy}*}

theory Topology_ZF_examples imports Topology_ZF Cardinal_ZF

begin

text{*
  This theory deals with some concrete examples of topologies.
  *}

section{*CoCardinal Topology of a set $X$*}

subsection{*CoCardinal topology is a topology.*}

text{*The collection of subsets of a set whose complement
is strictly bounded by a cardinal is a topology given some assumptions
on the cardinal.*}

definition Cocardinal ("CoCardinal _ _" 50) where
"CoCardinal X T ≡ {F∈Pow(X). X-F \<prec> T}∪ {0}"

text{*For any set and any infinite cardinal; we prove that
@{text "CoCardinal X Q"} forms a topology. The proof is done
with an infinite cardinal, but it is obvious that the set @{text "Q"}
can be any set equipollent with an infinite cardinal.
It is a topology also if the set where the topology is defined is
too small or the cardinal too large; in this case, as it is later proved the topology
is a discrete topology. And the last case corresponds with @{prop "Q=1"} which translates
in the indiscrete topology.*}

lemma CoCar_is_topology:
  assumes "InfCard (Q)"
  shows "(CoCardinal X Q) {is a topology}"
proof-
  let ?T="(CoCardinal X Q)"
  {
    fix M
    assume A:"M∈Pow(?T)"
    hence "M⊆?T" by auto
    then have "M⊆Pow(X)" using Cocardinal_def by auto
    then have "\<Union>M∈Pow(X)" by auto
    moreover
    {
      assume B:"M=0"
      then have "\<Union>M∈?T" using Cocardinal_def by auto
    }
    moreover
    {
      assume B:"M={0}"
      then have "\<Union>M∈?T" using Cocardinal_def by auto
    }
    moreover
    {
      assume B:"M ≠0" "M≠{0}"
      from B obtain T where C:"T∈M" and "T≠0" by auto
      with A have D:"X-T \<prec> (Q)" using Cocardinal_def by auto
      from C have "X-\<Union>M⊆X-T" by blast
      with D have "X-\<Union>M\<prec> (Q)" using subset_imp_lepoll lesspoll_trans1 by blast
    }
    ultimately have "\<Union>M∈?T" using Cocardinal_def by auto
  }
  moreover
  {
    fix U and V
    assume "U∈?T" and "V∈?T"
    hence A:"U=0 ∨ (U∈Pow(X) ∧ X-U\<prec> (Q))" and
      B:"V=0 ∨ (V∈Pow(X) ∧ X-V\<prec> (Q))" using Cocardinal_def by auto
    hence D:"U∈Pow(X)""V∈Pow(X)" by auto
    have C:"X-(U ∩ V)=(X-U)∪(X-V)" by fast
    with A B C have "U∩V=0∨(U∩V∈Pow(X) ∧ X-(U ∩ V)\<prec> (Q))" using less_less_imp_un_less assms
      by auto
    hence "U∩V∈?T" using Cocardinal_def by auto
  }
  ultimately show ?thesis using IsATopology_def by auto
qed

theorem topology0_CoCardinal:
  assumes "InfCard(T)"
  shows "topology0(CoCardinal X T)"
  using topology0_def CoCar_is_topology assms by auto

text{*It can also be proven that, if @{text "CoCardinal X T"} is a topology,
@{prop "X≠0"}, @{prop "Card(T)"} and @{prop "T≠0"}; then @{text "T"} is an infinite cardinal, @{prop "X\<prec>T"}
or @{text "T=1"}.
It follows from the fact that the union of two closed sets is closed. *}

text{*Choosing the appropriate cardinals, the cofinite and the cocountable topologies
are obtained.*}

text{*The cofinite topology is a very special topology because is extremely
related to the separation axiom $T_1$. It also appears naturally 
in algebraic geometry.*}

definition
  Cofinite ("CoFinite _" 90) where
  "CoFinite X ≡ CoCardinal X nat"

definition
  Cocountable ("CoCountable _" 90) where
  "CoCountable X ≡ CoCardinal X csucc(nat)"

subsection{*Total set, Closed sets, Interior, Closure and Boundary*}

text{*There are several assertions that can be done to the
 @{text "CoCardinal X T"} topology. In each case, we will not assume 
sufficient conditions for  @{text "CoCardinal X T"} to be a topology, but
they will be enough to do the calculations in every posible case.*}

text{*The topology is defined in the set $X$*}

lemma union_cocardinal:
  assumes "T≠0"
  shows "\<Union> (CoCardinal X T)=X"
proof-
  have X:"X-X=0" by auto
  have "0 \<lesssim> 0" by auto
  with assms have "0\<prec>1""1 \<lesssim>T" using not_0_is_lepoll_1 lepoll_imp_lesspoll_succ by auto
  then have "0\<prec>T" using lesspoll_trans2  by auto
  with X have "(X-X)\<prec>T" by auto
  then have "X∈(CoCardinal X T)" using Cocardinal_def by auto
  hence "X⊆\<Union> (CoCardinal X T)" by blast
  then show  "\<Union> (CoCardinal X T)=X" using Cocardinal_def by auto
qed

text{*The closed sets are the small subsets of $X$ and $X$ itself.*}

lemma closed_sets_cocardinal:
  assumes "T≠0"
  shows "D {is closed in} (CoCardinal X T) <-> (D∈Pow(X) & D\<prec>T)∨ D=X"
proof-
  {
    assume A:"D ⊆ X" "X - D ∈ (CoCardinal X T) "" D ≠ X"
    from A(1,3) have "X-(X-D)=D" "X-D≠0" by (safe,blast+)
    with A(2) have "D\<prec>T" using Cocardinal_def by simp
  }
  with assms have "D {is closed in} (CoCardinal X T) --> (D∈Pow(X) & D\<prec>T)∨ D=X" using IsClosed_def
    union_cocardinal by auto
  moreover
  {
    assume A:"D \<prec> T""D ⊆ X"
    from A(2) have "X-(X-D)=D" by blast
    with A(1) have "X-(X-D)\<prec> T" by auto
    then have "X-D∈ (CoCardinal X T)" using Cocardinal_def by auto
  }
  with assms have "(D∈Pow(X) & D\<prec>T)--> D {is closed in} (CoCardinal X T)" using union_cocardinal
    IsClosed_def by auto
  moreover
  have "X-X=0" by auto
  then have "X-X∈ (CoCardinal X T)"using Cocardinal_def by auto
  with assms have "X{is closed in} (CoCardinal X T)" using union_cocardinal
    IsClosed_def by auto
  ultimately show ?thesis by auto
qed

text{*The interior of a set is itself if it is open or @{text "0"} if
it isn't open.*}

lemma interior_set_cocardinal:
  assumes noC: "T≠0" and "A⊆X"
  shows "Interior(A,(CoCardinal X T))= (if ((X-A) \<prec> T) then A else 0)"
proof-
  from assms(2) have dif_dif:"X-(X-A)=A" by blast
  {
    assume "(X-A) \<prec> T"
    then have "(X-A)∈Pow(X) &(X-A) \<prec> T" by auto
    with noC have "(X-A) {is closed in} (CoCardinal X T)" using closed_sets_cocardinal
      by auto
    with noC have "X-(X-A)∈(CoCardinal X T)" using IsClosed_def union_cocardinal
      by auto
    with dif_dif have "A∈(CoCardinal X T)" by auto
    hence "A∈{U∈(CoCardinal X T). U ⊆ A}" by auto
    hence a1:"A⊆\<Union>{U∈(CoCardinal X T). U ⊆ A}" by auto
    have a2:"\<Union>{U∈(CoCardinal X T). U ⊆ A}⊆A" by blast
    from a1 a2 have "Interior(A,(CoCardinal X T))=A" using Interior_def by auto}
  moreover
  {
    assume as:"~((X-A) \<prec> T)"
    {
      fix U
      assume "U ⊆A"
      hence "X-A ⊆ X-U" by blast
      then have Q:"X-A \<lesssim> X-U" using subset_imp_lepoll by auto
      {
        assume "X-U\<prec> T"
        with Q have "X-A\<prec> T" using lesspoll_trans1 by auto
        with as have "False"  by auto
      }
      hence "~((X-U) \<prec> T)" by auto
      then have "U∉(CoCardinal X T)∨U=0" using Cocardinal_def by auto
    }
    hence "{U∈(CoCardinal X T). U ⊆ A}⊆{0}"  by blast
    then have "Interior(A,(CoCardinal X T))=0" using Interior_def by auto
  }
  ultimately show ?thesis by auto
qed

text{*$X$ is a closed set that contains $A$.
This lemma is necessary because we cannot
use the lemmas proven in the @{text "topology0"} context since
@{prop "T≠0"} is too weak for
 @{text "CoCardinal X T"} to be a topology.*}

lemma X_closedcov_cocardinal:
  assumes "T≠0""A⊆X"
  shows "X∈ClosedCovers(A,(CoCardinal X T))" using ClosedCovers_def
  using union_cocardinal closed_sets_cocardinal assms by auto

text{*The closure of a set is itself if it is closed or @{text "X"} if
it isn't closed.*}

lemma closure_set_cocardinal:
  assumes "T≠0""A⊆X"
  shows "Closure(A,(CoCardinal X T))=(if (A \<prec> T) then A else X)"
proof-
  {
    assume "A \<prec> T"
    with assms have "A {is closed in} (CoCardinal X T)" using closed_sets_cocardinal by auto
    with assms(2) have "A∈ {D ∈ Pow(X). D {is closed in} (CoCardinal X T) ∧ A⊆D}" by auto
    with assms(1) have S:"A∈ClosedCovers(A,(CoCardinal X T))" using ClosedCovers_def
      using union_cocardinal by auto
    hence l1:"\<Inter>ClosedCovers(A,(CoCardinal X T))⊆A" by blast
    from S have l2:"A⊆\<Inter>ClosedCovers(A,(CoCardinal X T))" 
        using ClosedCovers_def[where T="CoCardinal X T" and A="A"] by auto
    from l1 l2 have "Closure(A,(CoCardinal X T))=A" using Closure_def
      by auto
  }
  moreover
  {
    assume as:"¬ A \<prec> T"
    {
      fix U
      assume "A⊆U"
      then have Q:"A \<lesssim> U" using subset_imp_lepoll by auto
      {
        assume "U\<prec> T"
        with Q have "A\<prec> T" using lesspoll_trans1 by auto
        with as have "False" by auto
      }
      hence "¬ U \<prec> T" by auto
      with assms(1) have "¬(U {is closed in} (CoCardinal X T)) ∨ U=X" using closed_sets_cocardinal
      by auto
    }
    with assms(1) have "∀U∈Pow(X). U{is closed in}(CoCardinal X T)∧A⊆U-->U=X"
      by auto
    with assms(1) have "ClosedCovers(A,(CoCardinal X T))⊆{X}" 
      using union_cocardinal using ClosedCovers_def by auto
    with assms have "ClosedCovers(A,(CoCardinal X T))={X}" using X_closedcov_cocardinal
      by auto
    then have " Closure(A, CoCardinal X T) = X " using Closure_def by auto
  }
  ultimately show ?thesis by auto
qed

text{*The boundary of a set is @{text "0"} if $A$ and $X-A$ are closed,
 @{text "X"} if not $A$ neither $X-A$ are closed and; if only one is closed,
then the closed one is its boundary.*}

lemma boundary_cocardinal:
  assumes "T≠0""A ⊆X"
  shows "Boundary(A,(CoCardinal X T))=(if A\<prec> T then (if  (X-A)\<prec> T then 0 else A) else (if  (X-A)\<prec> T then X-A else X))"
proof-
  {
    assume AS:"A\<prec> T""X-A\<prec> T"
    from AS(2) assms have "Closure(X-A,(CoCardinal X T))=X-A" using closure_set_cocardinal[where A="X-A" and T="T" and X="X"] by auto
    moreover
    from AS(1) assms have "Closure(A,(CoCardinal X T))=A"
      using closure_set_cocardinal by auto
    with calculation assms(1) have "Boundary(A,(CoCardinal X T))=0"using Boundary_def using
      union_cocardinal by auto
  }
  moreover
  {
    assume AS:"~(A\<prec> T)""X-A\<prec> T"
    from AS(2) assms have "Closure(X-A,(CoCardinal X T))=X-A" using closure_set_cocardinal[where A="X-A" and T="T" and X="X"] by auto
    moreover
    from AS(1) assms have "Closure(A,(CoCardinal X T))=X"
      using closure_set_cocardinal by auto
    with calculation assms(1)  have "Boundary(A,(CoCardinal X T))=X-A" using Boundary_def
      union_cocardinal by auto
  }
  moreover
  {
    assume AS:"~(A\<prec> T)""~(X-A\<prec> T)"
    from AS(2) assms have "Closure(X-A,(CoCardinal X T))=X" using closure_set_cocardinal[where A="X-A" and T="T" and X="X"] by auto
    moreover
    from AS(1) assms have "Closure(A,(CoCardinal X T))=X"
      using closure_set_cocardinal by auto
    with calculation assms(1)  have "Boundary(A,(CoCardinal X T))=X"using Boundary_def
      union_cocardinal by auto
  }
  moreover
  {
    assume AS:"A\<prec> T""~(X-A\<prec> T)"
    from AS(2) assms have "Closure(X-A,(CoCardinal X T))=X" using closure_set_cocardinal[where A="X-A" and T="T" and X="X"] by auto
    moreover
    from AS(1) assms have "Closure(A,(CoCardinal X T))=A"
      using closure_set_cocardinal by auto
    with calculation assms have "Boundary(A,(CoCardinal X T))=A" using Boundary_def
      union_cocardinal by auto
  }
  ultimately show ?thesis by auto
qed

subsection{*Special cases and subspaces*}

text{*If the set is too small or the cardinal too large, then the topology
is just the discrete topology.*}

lemma discrete_cocardinal:
  assumes "X\<prec> T"
  shows "(CoCardinal X T)=(Pow (X))"
proof
  {
    fix U
    assume "U∈(CoCardinal X T)"
    then have "U∈Pow (X)" using Cocardinal_def by auto
  }
  then show "(CoCardinal X T)⊆(Pow (X))" by auto
  {
    fix U
    assume A:"U∈Pow(X)"
    then have "X-U ⊆ X" by auto
    then have "X-U \<lesssim>X" using subset_imp_lepoll by auto
    then have "X-U\<prec> T" using lesspoll_trans1 assms by auto
    with A have "U∈(CoCardinal X T)" using Cocardinal_def
      by auto
  }
  then show "Pow(X)⊆(CoCardinal X T)" by auto
qed

text{*If the cardinal is taken as @{prop "T=1"} then the topology
is indiscrete.*}

lemma indiscrete_cocardinal:
  shows "(CoCardinal X 1)={0,X}"
proof
  {
    fix Q
    assume "Q∈(CoCardinal X 1)"
    then have "Q∈Pow(X)""Q=0∨X-Q\<prec>1" using Cocardinal_def by auto
    then have "Q∈Pow(X)""Q=0∨X-Q=0" using lesspoll_succ_iff lepoll_0_iff by (safe,blast)
    then have "Q=0∨Q=X" by blast
  }
  then show "(CoCardinal X 1) ⊆ {0, X}" by auto
  have "0∈(CoCardinal X 1)" using Cocardinal_def by auto
  moreover
  have "0\<prec>1""X-X=0" using lesspoll_succ_iff by auto
  then have "X∈(CoCardinal X 1)" using Cocardinal_def by auto
  ultimately show "{0, X} ⊆ (CoCardinal X 1) " by auto
qed

text{*The topological subspaces of the @{text "CoCardinal X T"} topology
are also CoCardinal topologies.*}

lemma subspace_cocardinal:
  shows "(CoCardinal X T) {restricted to} Y=(CoCardinal (Y ∩ X) T)"
proof
  {
    fix M
    assume "M∈((CoCardinal X T) {restricted to} Y)"
    then obtain A where A1:"A:(CoCardinal X T)" "M=Y ∩ A" using RestrictedTo_def by auto
    then have "M∈Pow(X ∩ Y)" using Cocardinal_def by auto
    moreover
    from A1 have "(Y ∩ X)-M=(Y ∩ X)-A" using Cocardinal_def by auto
    have "(Y ∩ X)-A ⊆ X-A" by blast
    with `(Y ∩ X)-M=(Y ∩ X)-A` have "(Y ∩ X)-M⊆ X-A" by auto
    then have "(Y ∩ X)-M \<lesssim> X-A" using subset_imp_lepoll by auto
    with A1 have "(Y ∩ X)-M \<prec> T ∨ M=0" using lesspoll_trans1 using Cocardinal_def
      by (cases "A=0",simp,cases "Y ∩ A=0",simp+)
    ultimately have "M∈(CoCardinal (Y ∩ X) T)" using Cocardinal_def
      by auto
  }
  then show "(CoCardinal X T) {restricted to} Y ⊆(CoCardinal (Y ∩ X) T)" by auto
  {
    fix M
    let ?A="M ∪ (X-Y)"
    assume A:"M∈(CoCardinal (Y ∩ X) T)"
    {
      assume "M=0"
      hence "M=0 ∩ Y" by auto
      then have "M∈(CoCardinal X T) {restricted to} Y" using RestrictedTo_def
        Cocardinal_def by auto
    }
    moreover
    {
      assume AS:"M≠0"
      from A AS have A1:"(M∈Pow(Y ∩ X) ∧ (Y ∩ X)-M\<prec> T)" using Cocardinal_def by auto
      hence "?A∈Pow(X)" by blast
      moreover
      have "X-?A=(Y ∩ X)-M" by blast
      with A1 have "X-?A\<prec> T" by auto
      ultimately have "?A∈(CoCardinal X T)" using Cocardinal_def by auto
      then have AT:"Y ∩ ?A∈(CoCardinal X T) {restricted to} Y" using RestrictedTo_def
        by auto
      have "Y ∩ ?A=Y ∩ M" by blast
      also with A1 have "…=M" by auto
      finally have "Y ∩ ?A=M".
      with AT have "M∈(CoCardinal X T) {restricted to} Y"
        by auto
    }
    ultimately have "M∈(CoCardinal X T) {restricted to} Y" by auto
  }
  then show "(CoCardinal (Y ∩ X) T) ⊆ (CoCardinal X T) {restricted to} Y" by auto
qed

section{*Excluded Set Topology*}

text{*In this seccion, we consider all the subsets of a set
which have empty intersection with a fixed set.*}

subsection{*Excluded set topology is a topology.*}

definition
   ExcludedSet ("ExcludedSet _ _" 50) where
   "ExcludedSet X U ≡ {F∈Pow(X). U ∩ F=0}∪ {X}"

text{*For any set; we prove that
@{text "ExcludedSet X Q"} forms a topology.*}

theorem excludedset_is_topology:
  shows "(ExcludedSet X Q) {is a topology}"
proof-
  {
    fix M
    assume "M∈Pow(ExcludedSet X Q)"
    then have A:"M⊆{F∈Pow(X). Q ∩ F=0}∪ {X}" using ExcludedSet_def by auto
    hence "\<Union>M∈Pow(X)" by auto
    moreover
    {
      have B:"Q ∩\<Union>M=\<Union>{Q ∩T. T∈M}" by auto
      {
        assume "X∉M"
        with A have "M⊆{F∈Pow(X). Q ∩ F=0}" by auto
        with B have "Q ∩ \<Union>M=0" by auto
      }
      moreover
      {
        assume "X∈M"
        with A have "\<Union>M=X" by auto
      }
      ultimately have  "Q ∩ \<Union>M=0 ∨ \<Union>M=X" by auto
    }
    ultimately have "\<Union>M∈(ExcludedSet X Q)" using ExcludedSet_def by auto
  }
  moreover
  {
    fix U V
    assume "U∈(ExcludedSet X Q)" "V∈(ExcludedSet X Q)"
    then have "U∈Pow(X)""V∈Pow(X)""U=X∨ U ∩ Q=0""V=X∨ V ∩ Q=0" using ExcludedSet_def by auto
    hence "U∈Pow(X)""V∈Pow(X)""(U ∩ V)=X ∨ Q∩(U ∩ V)=0" by auto
    then have "(U ∩ V)∈(ExcludedSet X Q)" using ExcludedSet_def by auto
  }
  ultimately show ?thesis using IsATopology_def by auto
qed

theorem topology0_excludedset:
  shows "topology0(ExcludedSet X T)"
  using topology0_def excludedset_is_topology by auto

text{*Choosing a singleton set, it is considered a point excluded
topology.*}

definition
  ExcludedPoint ("ExcludedPoint _ _" 90) where
  "ExcludedPoint X p≡ ExcludedSet X {p}"

subsection{*Total set, Closed sets, Interior, Closure and Boundary*}

text{*The topology is defined in the set $X$*}

lemma union_excludedset:
  shows "\<Union> (ExcludedSet X T)=X"
proof-
  have "X∈(ExcludedSet X T)" using ExcludedSet_def by auto
  then show ?thesis using ExcludedSet_def by auto
qed

text{*The closed sets are those which contain the set @{text "(X ∩ T)"} and @{text "0"}.*}

lemma closed_sets_excludedset:
  shows "D {is closed in} (ExcludedSet X T) <-> (D∈Pow(X) & (X ∩ T) ⊆D)∨ D=0"
proof-
  {
    fix x
    assume A:"D ⊆ X" "X - D ∈ (ExcludedSet X T) "" D ≠ 0""x:T""x:X"
    from A(1) have B:"X-(X-D)=D" by auto
    from A(2) have "T∩(X-D)=0∨ X-D=X" using ExcludedSet_def by auto
    hence "T∩(X-D)=0∨ X-(X-D)=X-X" by auto
    with B have "T∩(X-D)=0∨ D=X-X" by auto
    hence "T∩(X-D)=0∨ D=0" by auto
    with A(3) have "T∩(X-D)=0" by auto
    with A(4) have "x∉X-D" by auto
    with A(5) have "x∈D" by auto
  }
  moreover
  {
    assume A:"X∩T⊆D""D⊆X"
    from A(1) have "X-D⊆X-(X∩T)" by auto
    also have "…=X-T" by auto
    finally have "T∩(X-D)=0" by auto
    moreover
    have "X-D∈Pow(X)" by auto
    ultimately have "X-D∈(ExcludedSet X T)" using ExcludedSet_def by auto
  }
  ultimately show ?thesis using IsClosed_def union_excludedset
    ExcludedSet_def by auto
qed

text{*The interior of a set is itself if it is @{text "X"} or
the difference with the set @{text"T"}*}

lemma interior_set_excludedset:
  assumes "A⊆X"
  shows "Interior(A,(ExcludedSet X T))= (if A=X then X else A-T)"
proof-
  {
    assume A:"A≠X"
    from assms have "A-T∈(ExcludedSet X T)" using ExcludedSet_def by auto
    then have "A-T⊆Interior(A,(ExcludedSet X T))"
    using Interior_def by auto
    moreover
    {
      fix U
      assume "U∈(ExcludedSet X T)""U⊆A"
      then have "T∩U=0 ∨ U=X""U⊆A" using ExcludedSet_def by auto
      with A assms have "T∩U=0""U⊆A" by auto
      then have "U-T=U""U-T⊆A-T" by (safe,blast+)
      then have "U⊆A-T" by auto
    }
    then have "Interior(A,(ExcludedSet X T))⊆A-T" using Interior_def by auto
    ultimately have "Interior(A,(ExcludedSet X T))=A-T" by auto
  }
  moreover
  have "X∈(ExcludedSet X T)" using ExcludedSet_def
  union_excludedset by auto
  then have "Interior(X,(ExcludedSet X T))=X" using topology0.Top_2_L3
  topology0_excludedset by auto
  ultimately show ?thesis by auto
qed

text{*The closure of a set is itself if it is @{text "0"} or
the union with @{text "T"}.*}

lemma closure_set_excludedset:
  assumes "A⊆X"
  shows "Closure(A,(ExcludedSet X T))=(if A=0 then 0 else A ∪(X∩ T))"
proof-
  have "0∈ClosedCovers(0,(ExcludedSet X T))" using ClosedCovers_def
    closed_sets_excludedset by auto
  then have "Closure(0,(ExcludedSet X T))⊆0" using Closure_def by auto
  hence "Closure(0,(ExcludedSet X T))=0" by blast
  moreover
  {
    assume A:"A≠0"
    then have "(A ∪(X∩ T)) {is closed in} (ExcludedSet X T)" 
      using closed_sets_excludedset[of "A ∪(X∩ T)"] assms A 
      by blast
    then have "(A ∪(X∩ T))∈ {D ∈ Pow(X). D {is closed in} (ExcludedSet X T) ∧ A⊆D}"
    using assms by auto
    then have "(A ∪(X∩ T))∈ClosedCovers(A,(ExcludedSet X T))" unfolding ClosedCovers_def
    using union_excludedset by auto
    then have l1:"\<Inter>ClosedCovers(A,(ExcludedSet X T))⊆(A ∪(X∩ T))" by blast
    {
      fix U
      assume "U∈ClosedCovers(A,(ExcludedSet X T))"
      then have "U{is closed in}(ExcludedSet X T)""A⊆U" using ClosedCovers_def
       union_excludedset by auto
      then have "U=0∨(X∩T)⊆U""A⊆U" using closed_sets_excludedset
       by auto
      then have "(X∩T)⊆U""A⊆U" using A by auto
      then have "(X∩T)∪A⊆U" by auto
    }
    then have "(A ∪(X∩ T))⊆\<Inter>ClosedCovers(A,(ExcludedSet X T))" using topology0.Top_3_L3
      topology0_excludedset union_excludedset assms by auto
    with l1 have "\<Inter>ClosedCovers(A,(ExcludedSet X T))=(A ∪(X∩ T))" by auto
    then have "Closure(A, ExcludedSet X T) = (A ∪(X∩ T)) "
    using Closure_def by auto
  }
  ultimately show ?thesis by auto
qed

text{*The boundary of a set is @{text "0"} if $A$ is @{text "X"}
or @{text "0"}, and @{text "X∩T"} in other case.*}

lemma boundary_excludedset:
  assumes "A ⊆X"
  shows "Boundary(A,(ExcludedSet X T))=(if A=0∨A=X then 0 else X∩T)"
proof-
  {
    have "Closure(0,(ExcludedSet X T))=0""Closure(X - 0,(ExcludedSet X T))=X"
    using closure_set_excludedset by auto
    then have "Boundary(0,(ExcludedSet X T))=0"using Boundary_def using
      union_excludedset assms by auto
  }
  moreover
  {
    have "X-X=0" by blast
    then have "Closure(X,(ExcludedSet X T))=X""Closure(X-X,(ExcludedSet X T))=0"
    using closure_set_excludedset by auto
    then have "Boundary(X,(ExcludedSet X T))=0"unfolding Boundary_def using
      union_excludedset by auto
  }
  moreover
  {
    assume AS:"(A≠0)∧(A≠X)"
    then have "(A≠0)""(X-A≠0)" using assms by (safe,blast)
    then have "Closure(A,(ExcludedSet X T))=A ∪ (X∩T)""Closure(X-A,(ExcludedSet X T))=(X-A) ∪ (X∩T)"
    using closure_set_excludedset[where A="A" and X="X"] assms closure_set_excludedset[where A="X-A"
      and X="X"] by auto
    then have "Boundary(A,(ExcludedSet X T))=X∩T" unfolding Boundary_def using
      union_excludedset by auto
  }
  ultimately show ?thesis by auto
qed

subsection{*Special cases and subspaces*}

text{*The topology is equal in the sets @{text "T"} and  
@{text "X∩T"}.*}

lemma smaller_excludedset:
  shows "(ExcludedSet X T)=(ExcludedSet X (X∩T))"
  using ExcludedSet_def by (simp,blast)

text{*If the set which is excluded is disjoint with @{text "X"},
then the topology is discrete.*}

lemma empty_excludedset:
  assumes "T∩X=0"
  shows "(ExcludedSet X T)=Pow(X)"
  using smaller_excludedset assms ExcludedSet_def by (simp,blast)

text{*The topological subspaces of the @{text "ExcludedSet X T"} topology
are also ExcludedSet topologies.*}

lemma subspace_excludedset:
  shows "(ExcludedSet X T) {restricted to} Y=(ExcludedSet (Y ∩ X) T)"
proof
  {
    fix M
    assume "M∈((ExcludedSet X T) {restricted to} Y)"
    then obtain A where A1:"A:(ExcludedSet X T)" "M=Y ∩ A" unfolding RestrictedTo_def by auto
    then have "M∈Pow(X ∩ Y)" unfolding ExcludedSet_def by auto
    moreover
    from A1 have "T∩M=0∨M=Y∩X" unfolding ExcludedSet_def by blast
    ultimately have "M∈(ExcludedSet (Y ∩ X) T)" unfolding ExcludedSet_def
      by auto
  }
  then show "(ExcludedSet X T) {restricted to} Y ⊆(ExcludedSet (Y ∩ X) T)" by auto
  {
    fix M
    let ?A="M ∪ ((X∩Y-T)-Y)"
    assume A:"M∈(ExcludedSet (Y ∩ X) T)"
    {
      assume "M=Y ∩ X"
      then have "M∈(ExcludedSet X T) {restricted to} Y" unfolding RestrictedTo_def
        ExcludedSet_def by auto
    }
    moreover
    {
      assume AS:"M≠Y ∩ X"
      from A AS have A1:"(M∈Pow(Y ∩ X) ∧ T∩M=0)" unfolding ExcludedSet_def by auto
      then have "?A∈Pow(X)" by blast
      moreover
      have "T∩?A=T∩M" by blast
      with A1 have "T∩?A=0" by auto
      ultimately have "?A∈(ExcludedSet X T)" unfolding ExcludedSet_def by auto
      then have AT:"Y ∩ ?A∈(ExcludedSet X T) {restricted to} Y"unfolding RestrictedTo_def
        by auto
      have "Y ∩ ?A=Y ∩ M" by blast
      also have "…=M" using A1 by auto
      finally have "Y ∩ ?A=M".
      then have "M∈(ExcludedSet X T) {restricted to} Y" using AT
        by auto
    }
    ultimately have "M∈(ExcludedSet X T) {restricted to} Y" by auto
  }
  then show "(ExcludedSet (Y ∩ X) T) ⊆ (ExcludedSet X T) {restricted to} Y" by auto
qed

section{*Included Set Topology*}

text{*In this section we consider the subsets of a set which
contain a fixed set.*}

text{*The family defined in this section and the one in the previous section are
dual; meaning that the closed set of one are the open sets of the other.*}

subsection{*Included set topology is a topology.*}

definition 
  IncludedSet ("IncludedSet _ _" 50) where
  "IncludedSet X U ≡ {F∈Pow(X). U ⊆ F}∪ {0}"

text{*For any set; we prove that
@{text "IncludedSet X Q"} forms a topology.*}

theorem includedset_is_topology:
  shows "(IncludedSet X Q) {is a topology}"
proof-
  {
    fix M
    assume "M∈Pow(IncludedSet X Q)"
    then have A:"M⊆{F∈Pow(X). Q ⊆ F}∪ {0}" using IncludedSet_def by auto
    then have "\<Union>M∈Pow(X)" by auto
    moreover
    have"Q ⊆\<Union>M∨ \<Union>M=0" using A by blast
    ultimately have "\<Union>M∈(IncludedSet X Q)" using IncludedSet_def by auto
  }
  moreover
  {
    fix U V
    assume "U∈(IncludedSet X Q)" "V∈(IncludedSet X Q)"
    then have "U∈Pow(X)""V∈Pow(X)""U=0∨ Q⊆U""V=0∨ Q⊆V" using IncludedSet_def by auto
    then have "U∈Pow(X)""V∈Pow(X)""(U ∩ V)=0 ∨ Q⊆(U ∩ V)" by auto
    then have "(U ∩ V)∈(IncludedSet X Q)" using IncludedSet_def by auto
  }
  ultimately show ?thesis using IsATopology_def by auto
qed

theorem topology0_includedset:
  shows "topology0(IncludedSet X T)"
  using topology0_def includedset_is_topology by auto

text{*Choosing a singleton set, it is considered a point excluded
topology. In the following lemmas and theorems, when neccessary
it will be considered that @{prop "T≠0"} and @{prop "T⊆X"}.
Theese cases will appear in the special cases section.*}

definition
  IncludedPoint ("IncludedPoint _ _" 90) where
  "IncludedPoint X p≡ IncludedSet X {p}"

subsection{*Total set, Closed sets, Interior, Closure and Boundary*}

text{*The topology is defined in the set $X$.*}

lemma union_includedset:
  assumes "T⊆X "
  shows "\<Union> (IncludedSet X T)=X"
proof-
  from assms have "X∈(IncludedSet X T)" using IncludedSet_def by auto
  then show "\<Union> (IncludedSet X T)=X" using IncludedSet_def by auto
qed

text{*The closed sets are those which are disjoint with @{text "T"}
 and @{text "X"}.*}

lemma closed_sets_includedset:
  assumes "T⊆X"
  shows "D {is closed in} (IncludedSet X T) <-> (D∈Pow(X) & (D ∩ T)=0)∨ D=X"
proof-
  have "X-X=0" by blast
  then have "X-X∈(IncludedSet X T)" using IncludedSet_def by auto
  moreover
  {
    assume A:"D ⊆ X" "X - D ∈ (IncludedSet X T) "" D ≠ X"
    from A(2) have "T⊆(X-D)∨ X-D=0" using IncludedSet_def by auto
    with A(1) have "T⊆(X-D)∨ D=X" by blast 
    with A(3) have "T⊆(X-D)" by auto
    hence "D∩T=0" by blast
  }
  moreover
  {
    assume A:"D∩T=0""D⊆X"
    from A(1) assms have "T⊆(X-D)" by blast
    then have "X-D∈(IncludedSet X T)" using IncludedSet_def by auto
  }
  ultimately show ?thesis using IsClosed_def union_includedset assms by auto
qed

text{*The interior of a set is itself if it is open or
 @{text"0"} if it isn't.*}

lemma interior_set_includedset:
  assumes "A⊆X"
  shows "Interior(A,(IncludedSet X T))= (if T⊆A then A else 0)"
proof-
  {
    fix x
    assume A:"Interior(A, IncludedSet X T) ≠ 0 ""x∈T"
    have "Interior(A,IncludedSet X T)∈(IncludedSet X T)" using
      topology0.Top_2_L2 topology0_includedset by auto
    with A(1) have "T⊆Interior(A, IncludedSet X T)" using IncludedSet_def
      by auto
    with A(2) have "x∈Interior(A, IncludedSet X T)" by auto
    then have "x∈A" using topology0.Top_2_L1 topology0_includedset by auto}
    moreover
  {
    assume "T⊆A"
    with assms have "A∈(IncludedSet X T)" using IncludedSet_def by auto
    then have "Interior(A,IncludedSet X T)=A" using topology0.Top_2_L3
      topology0_includedset by auto
  }
  ultimately show ?thesis by auto
qed

text{*The closure of a set is itself if it is closed or
 @{text "X"} if it isn't.*}

lemma closure_set_includedset:
  assumes "A⊆X""T⊆X"
  shows "Closure(A,(IncludedSet X T))= (if T∩A=0 then A else X)"
proof-
  {
    assume AS:"T∩A=0"
    then have "A {is closed in} (IncludedSet X T)" using closed_sets_includedset
      assms by auto
    with assms(1) have "Closure(A,(IncludedSet X T))=A" using topology0.Top_3_L8
      topology0_includedset union_includedset assms(2) by auto
  }
  moreover
  {
    assume AS:"T∩A≠0"
    have "X∈ClosedCovers(A,(IncludedSet X T))" using ClosedCovers_def
      closed_sets_includedset union_includedset assms by auto
    then have l1:"\<Inter>ClosedCovers(A,(IncludedSet X T))⊆X" using Closure_def
      by auto
    moreover
    {
      fix U
      assume "U∈ClosedCovers(A,(IncludedSet X T))"  
      then have "U{is closed in}(IncludedSet X T)""A⊆U" using ClosedCovers_def
        by auto
      then have "U=X∨(T∩U)=0""A⊆U" using closed_sets_includedset assms(2)
        by auto
      then have "U=X∨(T∩A)=0" by auto
      then have "U=X" using AS by auto
    }
    then have "X⊆\<Inter>ClosedCovers(A,(IncludedSet X T))" using topology0.Top_3_L3
      topology0_includedset union_includedset assms by auto
    ultimately have "\<Inter>ClosedCovers(A,(IncludedSet X T))=X" by auto
    then have "Closure(A, IncludedSet X T) = X "
      using Closure_def by auto
  }
  ultimately show ?thesis by auto
qed

text{*The boundary of a set is @{text "X-A"} if $A$ contains @{text "T"}
completely, is @{text "A"} if $X-A$ contains @{text "T"}
completely
and @{text "X"} if @{text "T"} is divided between the two sets.
The case where @{prop "T=0"} is considered as an special case.*}

lemma boundary_includedset:
  assumes "A ⊆X""T ⊆X""T≠0"
  shows "Boundary(A,(IncludedSet X T))=(if T⊆A then X-A else (if T∩A=0 then A else X))"
proof-
  {
    assume AS:"(T⊆A)"
    then have "T∩A≠0""T∩(X-A)=0" using assms(2,3) by (auto,blast)
    then have "Closure(A,(IncludedSet X T))=X""Closure(X-A,(IncludedSet X T))=(X-A)"
      using closure_set_includedset[where A="A" and X="X"and T="T"] assms(1,2) closure_set_includedset[where A="X-A"
      and X="X"and T="T"] by auto
    then have "Boundary(A,(IncludedSet X T))=X-A" unfolding Boundary_def using
      union_includedset assms(2) by auto
  }
  moreover
  {
    assume AS:"~(T⊆A)""T∩A=0"
    then have "T∩A=0""T∩(X-A)≠0" using assms(2) by (safe,blast+)
    then have "Closure(A,(IncludedSet X T))=A""Closure(X-A,(IncludedSet X T))=X"
      using closure_set_includedset[where A="A" and X="X"and T="T"] assms(1,2) closure_set_includedset[where A="X-A"
      and X="X"and T="T"] by auto
    then have "Boundary(A,(IncludedSet X T))=A" unfolding Boundary_def using
      union_includedset assms(1,2) by auto
  }
  moreover
  {
    assume AS:"~(T⊆A)""T∩A≠0"
    then have "T∩A≠0""T∩(X-A)≠0" using assms(2) by (safe,blast+)
    then have "Closure(A,(IncludedSet X T))=X""Closure(X-A,(IncludedSet X T))=X"
      using closure_set_includedset[where A="A" and X="X"and T="T"] assms(1,2) closure_set_includedset[where A="X-A"
      and X="X"and T="T"] by auto
    then have "Boundary(A,(IncludedSet X T))=X" unfolding Boundary_def using
      union_includedset assms(2) by auto
  }
  ultimately show ?thesis by auto
qed

subsection{*Special cases and subspaces*}

text{*The topology is discrete if @{prop "T=0"}*}

lemma smaller_includedset:
  shows "(IncludedSet X 0)=Pow(X)"
  using IncludedSet_def by (simp,blast)

text{*If the set which is included is not a subset of @{text "X"},
then the topology is trivial.*}

lemma empty_includedset:
  assumes "~(T⊆X)"
  shows "(IncludedSet X T)={0}"
  using assms IncludedSet_def by (simp,blast)

text{*The topological subspaces of the @{text "IncludedSet X T"} topology
are also IncludedSet topologies. The trivial case does not fit the idea
in the demonstration;
because if @{prop "Y⊆X"} then @{text "IncludedSet (Y ∩ X) (Y∩T)"}
is never trivial. There is no need of a separate proof because
the only subspace of the trivial topology is itself.*}

lemma subspace_includedset:
  assumes "T⊆X"
  shows "(IncludedSet X T) {restricted to} Y=(IncludedSet (Y ∩ X) (Y∩T))"
proof
  {
    fix M
    assume "M∈((IncludedSet X T) {restricted to} Y)"
    then obtain A where A1:"A:(IncludedSet X T)" "M=Y ∩ A" unfolding RestrictedTo_def by auto
    then have "M∈Pow(X ∩ Y)" unfolding IncludedSet_def by auto
    moreover
    from A1 have "Y∩T⊆M∨M=0" unfolding IncludedSet_def by blast
    ultimately have "M∈(IncludedSet (Y ∩ X) (Y∩T))" unfolding IncludedSet_def
      by auto
  }
  then show "(IncludedSet X T) {restricted to} Y ⊆(IncludedSet (Y ∩ X) (Y∩T))" by auto
  {
    fix M
    let ?A="M ∪ T"
    assume A:"M∈(IncludedSet (Y ∩ X) (Y∩T))"
    {
      assume "M=0"
      then have "M∈(IncludedSet X T) {restricted to} Y" unfolding RestrictedTo_def
        IncludedSet_def by auto
    }
    moreover
    {
      assume AS:"M≠0"
      from A AS have A1:"(M∈Pow(Y ∩ X) ∧ Y ∩T⊆M)" unfolding IncludedSet_def by auto
      then have "?A∈Pow(X)" using assms by blast
      moreover
      have "T⊆?A" by blast
      ultimately have "?A∈(IncludedSet X T)" unfolding IncludedSet_def by auto
      then have AT:"Y ∩ ?A∈(IncludedSet X T) {restricted to} Y"unfolding RestrictedTo_def
        by auto
      from A1 have "Y ∩ ?A=Y ∩ M" by blast
      also with A1 have "…=M" by auto
      finally have "Y ∩ ?A=M".
      with AT have "M∈(IncludedSet X T) {restricted to} Y"
        by auto
    }
    ultimately have "M∈(IncludedSet X T) {restricted to} Y" by auto
  }
  thus "(IncludedSet (Y ∩ X) (Y ∩ T)) ⊆ (IncludedSet X T) {restricted to} Y" by auto
qed

end