The Mizar article:

On \Tone\ Reflex of Topological Space

by
Adam Naumowicz, and
Mariusz \Lapinski

Received March 7, 1998

Copyright (c) 1998 Association of Mizar Users

MML identifier: T_1TOPSP
[ MML identifier index ]


environ

 vocabulary FUNCT_1, RELAT_1, PRE_TOPC, EQREL_1, BORSUK_1, TARSKI, BOOLE,
      SUBSET_1, SETFAM_1, PROB_1, URYSOHN1, ORDINAL2, T_1TOPSP;
 notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2,
      SETFAM_1, STRUCT_0, PRE_TOPC, EQREL_1, TOPS_2, URYSOHN1, BORSUK_1;
 constructors URYSOHN1, TOPS_2;
 clusters PRE_TOPC, BORSUK_1, STRUCT_0, FSM_1, FUNCT_1, RELSET_1, SUBSET_1,
      XBOOLE_0, EQREL_1, PARTFUN1;
 requirements BOOLE, SUBSET;
 definitions TARSKI, EQREL_1, PRE_TOPC, XBOOLE_0;
 theorems ZFMISC_1, PRE_TOPC, TARSKI, SETFAM_1, EQREL_1, URYSOHN1, BORSUK_1,
      FUNCT_2, FUNCT_1, TOPS_2, RELSET_1, RELAT_1, XBOOLE_0, XBOOLE_1;
 schemes FUNCT_1;

begin

reserve X for non empty set, x for Element of X, y,v,w for set;

canceled;

theorem Th2:
 for T being non empty TopSpace,
     A being non empty a_partition of the carrier of T,
     y being Subset of space A holds (Proj(A))"y = union y
      proof
       let T be non empty TopSpace;
       let A be non empty a_partition of the carrier of T;
       let y be Subset of space A;
       reconsider y as Subset of A by BORSUK_1:def 10;
         (Proj(A))"y = (proj A)"y by BORSUK_1:def 11
                      .= union y by BORSUK_1:31;
       hence thesis;
      end;

theorem Th3:
 for X being non empty set,
     S being a_partition of X,
     A being Subset of S holds (union S) \ (union A) = union (S \ A)
      proof
       let X be non empty set;
       let S be a_partition of X;
       let A be Subset of S;
       thus (union S) \ (union A) c= union (S \ A)
        proof
         let y be set;
         assume y in (union S) \ (union A);
         then A1:       y in (union S) & not y in (union A) by XBOOLE_0:def 4;
         then consider z being set such that
A2:       y in z & z in S by TARSKI:def 4;
           not z in A by A1,A2,TARSKI:def 4;
         then y in z & z in S \ A by A2,XBOOLE_0:def 4;
         hence thesis by TARSKI:def 4;
        end;
       thus union (S \ A) c= (union S) \ (union A)
        proof
         let y be set;
         assume y in union(S \ A);
         then consider z being set such that
A3:       y in z & z in S \ A by TARSKI:def 4;
A4:      y in z & z in S & not z in A by A3,XBOOLE_0:def 4;
         then A5:      y in union S by TARSKI:def 4;
         A6: now
          let w be set;
          assume
A7:        w in A;
          then w in S;
          then z misses w by A4,A7,EQREL_1:def 6;
          hence not y in w by A3,XBOOLE_0:3;
         end;
           now
          assume y in union A;
          then consider v being set such that
A8:        y in v & v in A by TARSKI:def 4;
          thus contradiction by A6,A8;
         end;
         hence thesis by A5,XBOOLE_0:def 4;
        end;
      end;

theorem Th4:
 for X being non empty set,A being Subset of X,
     S being a_partition of X holds A in S implies union(S \ {A}) = X \ A
      proof
       let X be non empty set;
       let A be Subset of X;
       let S be a_partition of X;
       assume
A1:     A in S;
         {A} c= S
       proof
        let y be set;
        assume y in {A};
        hence thesis by A1,TARSKI:def 1;
       end;
       then reconsider AA = {A} as Subset of S;
       thus union (S \ {A}) = (union S) \ (union AA) by Th3
                           .= X \ (union {A}) by EQREL_1:def 6
                           .= X \ A by ZFMISC_1:31;
      end;

theorem Th5:
 for T being non empty TopSpace,
     S being non empty a_partition of the carrier of T,
     A being Subset of space S,
     B being Subset of T holds
     B = union A implies (A is closed iff B is closed)
      proof
       let T be non empty TopSpace;
       let S be non empty a_partition of the carrier of T;
       let A be Subset of space S;
       let B be Subset of T;
       assume
A1:   B = union A;
       reconsider C = A as Subset of S by BORSUK_1:def 10;
A2:    [#](T) \ union A = (the carrier of T) \ union A by PRE_TOPC:def 3
                     .= (union S) \ (union C) by EQREL_1:def 6
                     .= union (S \ A) by Th3
                     .= union ((the carrier of space S) \ A)
                                          by BORSUK_1:def 10
                     .= union ([#](space S) \ A) by PRE_TOPC:def 3;
       thus A is closed implies B is closed
        proof
         assume A is closed;
then A3:       [#](space S) \ A is open by PRE_TOPC:def 6;
         reconsider om = [#](space S) \ A as Subset of S by BORSUK_1:def 10;
         om in the topology of space S by A3,PRE_TOPC:def 5;
         then [#](T) \ B in the topology of T by A1,A2,BORSUK_1:69;
         then [#](T) \ B is open by PRE_TOPC:def 5;
         hence thesis by PRE_TOPC:def 6;
        end;
       thus B is closed implies A is closed
        proof
         assume B is closed;
         then [#](T) \ B is open by PRE_TOPC:def 6;
         then A4:       [#](T) \ union A in the topology of T by A1,PRE_TOPC:
def 5;
         reconsider om = [#](space S) \ A as Subset of S by BORSUK_1:def 10;
           om in the topology of space S by A2,A4,BORSUK_1:69;
         then [#](space S) \ A is open by PRE_TOPC:def 5;
         hence thesis by PRE_TOPC:def 6;
        end;
      end;

:: Classes of partitions of a set

definition
 let X be non empty set,x be Element of X,S1 be a_partition of X;
 func EqClass(x,S1) -> Subset of X means :Def1:
 x in it & it in S1;

 existence
  proof
   consider EQR being Equivalence_Relation of X such that
A1: S1 = Class EQR by EQREL_1:43;
   take Class(EQR,x);
   thus x in Class(EQR,x) by EQREL_1:28;
   thus Class(EQR,x) in S1 by A1,EQREL_1:def 5;
  end;
 uniqueness
  proof
   let A,B be Subset of X; assume
A2: x in A & A in S1 & x in B & B in S1;
   then A meets B by XBOOLE_0:3;
   hence A = B by A2,EQREL_1:def 6;
  end;
end;

theorem Th6:
 for S1,S2 being a_partition of X st
 (for x being Element of X holds EqClass(x,S1) = EqClass(x,S2)) holds S1=S2
  proof
   let S1,S2 be a_partition of X;
   assume
A1: for x being Element of X holds EqClass(x,S1) = EqClass(x,S2);
     now let P be Subset of X;
    thus P in S1 implies P in S2
     proof
      assume
A2:    P in S1;
      then A3:    P is non empty by EQREL_1:def 6;
      consider x being Element of P;
        x in P by A3;
      then reconsider x as Element of X;
        P = EqClass(x,S1) by A2,A3,Def1;
      then P = EqClass(x,S2) by A1;
      hence P in S2 by Def1;
     end;
    thus P in S2 implies P in S1
     proof
      assume
A4:   P in S2;
then A5:   P <> {} by EQREL_1:def 6;
      consider x being Element of P;
        x in P by A5;
      then reconsider x as Element of X;
        P = EqClass(x,S2) by A4,A5,Def1;
      then P = EqClass(x,S1) by A1;
      hence P in S1 by Def1;
     end;
   end;
   hence thesis by SETFAM_1:44;
  end;

theorem Th7:
 for X being non empty set holds {X} is a_partition of X
  proof
   let X be non empty set;
     {X} c= bool X by ZFMISC_1:80;
   then reconsider A1 = {X} as Subset-Family of X by SETFAM_1:def 7;
A1: union A1 = X by ZFMISC_1:31;
  for A being Subset of X st A in A1 holds A <> {} & for B being
             Subset of X st B in A1 holds A = B or A misses B
    proof
     let A be Subset of X;
     assume
A2:  A in A1;
     hence A <> {} by TARSKI:def 1;
     let B be Subset of X;
     assume B in A1;
     then B = X by TARSKI:def 1;
     hence A = B or A misses B by A2,TARSKI:def 1;
    end;
   hence {X} is a_partition of X by A1,EQREL_1:def 6;
  end;

definition let X be set;
  mode Family-Class of X means :Def2:
   it c= bool bool X;
  existence;
end;

Lm1: for X being set holds {} is Family-Class of X
  proof
    let X be set;
      {} c= bool bool X by XBOOLE_1:2;
    hence thesis by Def2;
  end;

definition let X be set;
 let F be Family-Class of X;
 attr F is partition-membered means :Def3:
   for S being set st S in F holds S is a_partition of X;
end;

definition let X be set;
 cluster partition-membered Family-Class of X;
 existence
  proof
   reconsider E = {} as Family-Class of X by Lm1;
   take E;
   let S be set; assume S in E;
   hence thesis;
  end;
end;

definition let X be set;
  mode Part-Family of X is partition-membered Family-Class of X;
end;

reserve F for Part-Family of X;

definition
 let X be non empty set;
 cluster non empty a_partition of X;
 existence
  proof
   take {X};
   thus thesis by Th7;
 end;
end;

theorem Th8:
 for X being set, p being a_partition of X holds
  {p} is Part-Family of X
   proof
    let X be set;
    let p be a_partition of X;
    A1: {p} c= bool bool X
    proof
      let x be set; assume x in {p};
      then x is a_partition of X by TARSKI:def 1;
      hence thesis;
    end;
      for x be set st x in {p} holds x is a_partition of X by TARSKI:def 1;
    hence thesis by A1,Def2,Def3;
end;

definition
 let X be set;
 cluster non empty Part-Family of X;

 existence
  proof
   consider p being a_partition of X;
    reconsider P = {p} as Part-Family of X by Th8;
   take P;
   thus thesis;
  end;
end;

theorem Th9:
 for S1 being a_partition of X,
     x,y being Element of X holds
    EqClass(x,S1) meets EqClass(y,S1) implies EqClass(x,S1) = EqClass(y,S1)
     proof
      let S1 be a_partition of X;
      let x,y be Element of X;
      assume
A1:    EqClass(x,S1) meets EqClass(y,S1);
      consider EQR being Equivalence_Relation of X such that
A2:    S1 = Class EQR by EQREL_1:43;
A3:    Class(EQR,x) = EqClass(x,S1)
       proof
A4:     x in Class(EQR,x) by EQREL_1:28;
         Class(EQR,x) in S1 by A2,EQREL_1:def 5;
       hence thesis by A4,Def1;
      end;
        Class(EQR,y) = EqClass(y,S1)
       proof
A5:     y in Class(EQR,y) by EQREL_1:28;
         Class(EQR,y) in S1 by A2,EQREL_1:def 5;
       hence thesis by A5,Def1;
      end;
      hence thesis by A1,A3,EQREL_1:32;
     end;

Lm2:
 for A being set holds
  A in {EqClass(x,S) where S is a_partition of X : S in F} implies
  ex T being a_partition of X st T in F & A = EqClass(x,T)
   proof
    let A be set;
    assume
      A in {EqClass(x,S) where S is a_partition of X : S in F};
    then consider S being a_partition of X such that
A1:  A = EqClass(x,S) & S in F;
    take S;
    thus thesis by A1;
   end;

theorem Th10:
 for A being set,X being non empty set,S being a_partition of X holds
 A in S implies ex x being Element of X st A = EqClass(x,S)
  proof
   let A be set,X be non empty set,S be a_partition of X;
   assume
A1: A in S;
   then A is non empty by EQREL_1:def 6;
   then consider x being set such that
A2: x in A by XBOOLE_0:def 1;
   take x;
   thus thesis by A1,A2,Def1;
  end;

definition
 let X be non empty set,F be non empty Part-Family of X;
 func Intersection F -> non empty a_partition of X means :Def4:
 for x being Element of X holds EqClass(x,it)
  = meet{EqClass(x,S) where S is a_partition of X : S in F};
 uniqueness
  proof
   given S1,S2 being a_partition of X such that
A1: for x being Element of X holds
     EqClass(x,S1) = meet{EqClass(x,S) where S is a_partition of X : S in F}
     and
A2: for x being Element of X holds
     EqClass(x,S2) = meet{EqClass(x,S) where S is a_partition of X : S in F}
     and
A3: not S1 = S2;
     now let x be Element of X;
      EqClass(x,S1) =
         meet{EqClass(x,S) where S is a_partition of X : S in F} by A1;
    hence EqClass(x,S1) = EqClass(x,S2) by A2;
   end;
   hence contradiction by A3,Th6;
  end;
 existence
  proof
   thus ex G being non empty a_partition of X st
         for x being Element of X holds EqClass(x,G)
              = meet{EqClass(x,S) where S is a_partition of X : S in F}
    proof
     set G = {meet{EqClass(x,S) where S is a_partition of X : S in F}
                   where x is Element of X : not contradiction };
       G c= bool X
      proof
       let y be set;
       assume y in G;
       then consider x being Element of X such that
A4:    y = meet{EqClass(x,S) where S is a_partition of X : S in F};
         y c= X
        proof
         let e be set;
         assume
A5:      e in y;
         consider T being set such that
A6:      T in F by XBOOLE_0:def 1;
         reconsider T as a_partition of X by A6,Def3;
           EqClass(x,T) in {EqClass(x,S) where S is a_partition of X : S in F}
                   by A6;
         then consider f being set such that
A7:      f in {EqClass(x,S) where S is a_partition of X : S in F};
         consider S being a_partition of X such that
A8:      f = EqClass(x,S) & S in F by A7;
           e in EqClass(x,S) by A4,A5,A7,A8,SETFAM_1:def 1;
         hence e in X;
        end;
       hence y in bool X;
      end;
     then reconsider G as Subset-Family of X by SETFAM_1:def 7;
       G is a_partition of X
      proof
       per cases;
       case X <> {};
       thus union G = X
        proof
         thus union G c= X;
         thus X c= union G
          proof
           let a be set;
           assume a in X;
           then reconsider a1=a as Element of X;
           consider T being set such that
A9:        T in F by XBOOLE_0:def 1;
           reconsider T as a_partition of X by A9,Def3;
A10:        EqClass(a1,T) in
           {EqClass(a1,S) where S is a_partition of X : S in F} by A9;
             for T being set
           st T in {EqClass(a1,S) where S is a_partition of X : S in F} holds
           a1 in T
            proof
             let T be set;
             assume
               T in {EqClass(a1,S) where S is a_partition of X : S in F};
             then consider A being a_partition of X such that
A11:          T = EqClass(a1,A)
             and A in F;
             thus thesis by A11,Def1;
            end;
           then A12:        a in meet{EqClass(a1,S) where S is a_partition of X
: S in F}
                                       by A10,SETFAM_1:def 1;
             meet{EqClass(a1,S) where S is a_partition of X : S in F} in G;
           hence a in union G by A12,TARSKI:def 4;
          end;
        end;
       let A be Subset of X;
       assume
A13:   A in G;
       then consider x being Element of X such that
A14:    A = meet{EqClass(x,S) where S is a_partition of X : S in F};
       consider T being set such that
A15:    T in F by XBOOLE_0:def 1;
       reconsider T as a_partition of X by A15,Def3;
A16:    EqClass(x,T) in {EqClass(x,S) where S is a_partition of X : S in F}
                     by A15;
         for Y being set
       st Y in {EqClass(x,S) where S is a_partition of X : S in F} holds
       x in Y
        proof
         let Y be set;
         assume
           Y in {EqClass(x,S) where S is a_partition of X : S in F};
         then ex T being a_partition of X st Y = EqClass(x,T) & T in F;
         hence x in Y by Def1;
        end;
       hence A<>{} by A14,A16,SETFAM_1:def 1;
       consider x being Element of X such that
A17:    A = meet{EqClass(x,S) where S is a_partition of X : S in F} by A13;

       let B be Subset of X;
       assume B in G;
       then consider y being Element of X such that
A18:    B = meet{EqClass(y,S) where S is a_partition of X : S in F};
       thus A = B or A misses B
        proof
A19:      {EqClass(x,S) where S is a_partition of X : S in F} is non empty
           proof
            consider T being set such that
A20:         T in F by XBOOLE_0:def 1;
            reconsider T as a_partition of X by A20,Def3;
              EqClass(x,T) in
            {EqClass(x,S) where S is a_partition of X : S in F} by A20;
            hence thesis;
           end;
A21:      {EqClass(y,S) where S is a_partition of X : S in F} is non empty
           proof
            consider T being set such that
A22:         T in F by XBOOLE_0:def 1;
            reconsider T as a_partition of X by A22,Def3;
              EqClass(y,T) in
            {EqClass(y,S) where S is a_partition of X : S in F} by A22;
            hence thesis;
           end;
           now
         assume A meets B;
         then consider c being set such that
A23:      c in A & c in B by XBOOLE_0:3;
            c in meet{EqClass(x,S) where S is a_partition of X : S in F} /\
              meet{EqClass(y,S) where S is a_partition of X : S in F}
          by A17,A18,A23,XBOOLE_0:def 3;
         then A24: c in meet({EqClass(x,S) where S is a_partition of X : S in F
}
             \/ {EqClass(y,S) where S is a_partition of X : S in F})
                                                      by A19,A21,SETFAM_1:10;

A25:     now let T be a_partition of X;
          assume
            T in F;
          then EqClass(x,T) in {EqClass(x,S) where S is a_partition of X : S
in F};
          then EqClass(x,T) in {EqClass(x,S) where S is a_partition of X : S
in F}
        \/ {EqClass(y,S) where S is a_partition of X : S in F} by XBOOLE_0:def
2;
          hence c in EqClass(x,T) by A24,SETFAM_1:def 1;
         end;
         A26: now
          let T be a_partition of X;
          assume T in F;
          then EqClass(y,T) in {EqClass(y,S) where S is a_partition of X : S
in F};
          then EqClass(y,T) in {EqClass(x,S) where S is a_partition of X : S
in F}
        \/ {EqClass(y,S) where S is a_partition of X : S in F} by XBOOLE_0:def
2;
          hence c in EqClass(y,T) by A24,SETFAM_1:def 1;
         end;
A27:      for T being a_partition of X st T in F holds
         ex A being set st A in EqClass(x,T) & A in EqClass(y,T)
          proof
           let T be a_partition of X;
           assume
A28:        T in F;
           take c;
           thus thesis by A25,A26,A28;
          end;
A29:      for T being a_partition of X st T in F holds
         (EqClass(x,T)) meets (EqClass(y,T))
          proof
           let T be a_partition of X; assume T in F;
           then consider A being set such that
A30:        A in EqClass(x,T) & A in EqClass(y,T) by A27;
           thus thesis by A30,XBOOLE_0:3;
          end;
A31:       for T being a_partition of X st T in F holds
                             EqClass(x,T) = EqClass(y,T)
          proof
           let T be a_partition of X;
           assume T in F;
           then EqClass(x,T) meets EqClass(y,T) by A29;
           hence thesis by Th9;
          end;
            {EqClass(x,S) where S is a_partition of X : S in F} =
          {EqClass(y,S) where S is a_partition of X : S in F}
           proof
            thus {EqClass(x,S) where S is a_partition of X : S in F}
             c= {EqClass(y,S) where S is a_partition of X : S in F}
             proof
              let A be set;
              assume
                A in {EqClass(x,S) where S is a_partition of X : S in F};
              then consider T being a_partition of X such that
A32:           T in F &
              A = EqClass(x,T) by Lm2;
                A = EqClass(y,T) by A31,A32;
              hence thesis by A32;
             end;
            thus {EqClass(y,S) where S is a_partition of X : S in F}
             c= {EqClass(x,S) where S is a_partition of X : S in F}
             proof
              let A be set;
              assume A in {EqClass(y,S) where S is a_partition of X : S in F};
              then consider T being a_partition of X such that
A33:           T in F & A = EqClass(y,T) by Lm2;
                A = EqClass(x,T) by A31,A33;
              hence thesis by A33;
             end;
           end;
          hence thesis by A17,A18;
         end;
         hence thesis;
        end;

       case X = {};
       hence G = {};
      end;
     then reconsider G as non empty a_partition of X;
     take G;
       for x being Element of X holds EqClass(x,G)
         = meet{EqClass(x,S) where S is a_partition of X : S in F}
      proof
       let x be Element of X;
A34:     meet{EqClass(x,S) where S is a_partition of X : S in F} in G;
      x in meet{EqClass(x,S) where S is a_partition of X : S in F}
        proof
A35:       {EqClass(x,S) where S is a_partition of X : S in F} is non empty
           proof
            consider T being set such that
A36:          T in F by XBOOLE_0:def 1;
            reconsider T as a_partition of X by A36,Def3;
              EqClass(x,T) in
            {EqClass(x,S) where S is a_partition of X : S in F} by A36;
            hence thesis;
           end;
           now
          let Y be set;
          assume Y in {EqClass(x,S) where S is a_partition of X : S in F};
          then ex T being a_partition of X st
          Y = EqClass(x,T) & T in F;
          hence x in Y by Def1;
         end;
         hence thesis by A35,SETFAM_1:def 1;
        end;
       hence thesis by A34,Def1;
      end;
     hence thesis;
  end;
 end;
end;

:: Families of partitions of topological spaces

reserve T for non empty TopSpace;

theorem Th11:
 { A where A is a_partition of the carrier of T : A is closed } is
                                        Part-Family of the carrier of T
  proof
   set S = { A where A is a_partition of the carrier of T : A is closed };
   A1: S c= bool bool the carrier of T
   proof
     let B be set; assume B in S;
     then consider A being a_partition of the carrier of T such that
A2:   B = A and A is closed;
     thus thesis by A2;
   end;
     now let B be set;
   assume B in { A where A is a_partition of the carrier of T : A is closed };
   then consider A being a_partition of the carrier of T such that
A3: B = A and A is closed;
   thus B is a_partition of the carrier of T by A3;
   end;
   hence thesis by A1,Def2,Def3;
  end;

definition
 let T;
 func Closed_Partitions T -> non empty Part-Family of the carrier of T equals
  :Def5:
 { A where A is a_partition of the carrier of T : A is closed };
 coherence
  proof
   set F = { A where A is a_partition of the carrier of T : A is closed };
   reconsider ct = {the carrier of T} as
                            a_partition of the carrier of T by Th7;
     for A being Subset of T st A in ct holds A is closed
    proof
     let A be Subset of T; assume A in ct;
     then A = the carrier of T by TARSKI:def 1;
     then A = [#](T) by PRE_TOPC:def 3;
     hence thesis;
    end;
  then ct is closed by TOPS_2:def 2;
  then ct in F; hence thesis by Th11;
 end;
end;

:: T_1 reflex of a topological space

definition
 let T be non empty TopSpace;
 func T_1-reflex T -> TopSpace equals :Def6:
   space (Intersection Closed_Partitions T);
 correctness;
end;

definition
 let T;
 cluster T_1-reflex T -> strict non empty;
 coherence
  proof
     T_1-reflex T = space (Intersection Closed_Partitions T) by Def6;
   hence thesis;
 end;
end;

theorem Th12:
 for T being non empty TopSpace holds T_1-reflex T is being_T1
  proof
   let T be non empty TopSpace;
     now
    let p be Point of T_1-reflex T;
      T_1-reflex T = space (Intersection Closed_Partitions T) by Def6;
    then A1: the carrier of T_1-reflex T = Intersection Closed_Partitions T
                                    by BORSUK_1:def 10;
    then A2:  p in Intersection Closed_Partitions T;
    consider x being Element of T such that
A3:  p = EqClass(x,Intersection Closed_Partitions T) by A1,Th10;

    A4:  p = meet { EqClass(x,S) where S is a_partition of the carrier of T :
    S in Closed_Partitions T } by A3,Def4;
    reconsider q=p as Subset of T by A3;

A5:  { EqClass(x,S) where S is a_partition of the carrier of T :
               S in Closed_Partitions T } is non empty
     proof
      consider Y being set such that
A6:   Y in Closed_Partitions T by XBOOLE_0:def 1;
      reconsider Y as a_partition of the carrier of T by A6,Def3;
        EqClass(x,Y) in
      {EqClass(x,S) where S is a_partition of the carrier of T
      : S in Closed_Partitions T} by A6;
      hence thesis;
     end;
   { EqClass(x,S) where S is a_partition of the carrier of T :
               S in Closed_Partitions T } c= bool the carrier of T
     proof
      let Z be set;
      assume Z in { EqClass(x,S) where S is a_partition of the carrier of T
                  : S in Closed_Partitions T };
      then consider Y being a_partition of the carrier of T such that
A7:   Z = EqClass(x,Y) & Y in Closed_Partitions T;
      thus thesis by A7;
     end;
    then reconsider m = { EqClass(x,S) where S is a_partition of the carrier of
T :
    S in Closed_Partitions T } as non empty Subset-Family of T
      by A5,SETFAM_1:def 7;
    reconsider m as non empty Subset-Family of T;
   for A being Subset of T st A in m holds A is closed
     proof
      let A be Subset of T;
      assume A in m;
      then consider S being a_partition of the carrier of T such that
A8:    A = EqClass(x,S) & S in Closed_Partitions T;
        S in { B where B is a_partition of the carrier of T : B is closed }
                       by A8,Def5;
      then consider B being a_partition of the carrier of T such that
A9:    S = B & B is closed;
        A in S by A8,Def1;
      hence thesis by A9,TOPS_2:def 2;
     end;
    then q is closed by A4,PRE_TOPC:44;
    then [#](T) \ q is open by PRE_TOPC:def 6;
    then [#](T) \ p in the topology of T by PRE_TOPC:def 5;
    then (the carrier of T) \ p in the topology of T by PRE_TOPC:def 3;
    then A10:  union((Intersection Closed_Partitions T) \ {p}) in the topology
of T
                                         by A2,Th4;
    reconsider I = (Intersection Closed_Partitions T) \ {p} as
    Subset of (Intersection Closed_Partitions T) by XBOOLE_1:36;
A11:  I in {A where A is Subset of (Intersection Closed_Partitions T) :
    union A in the topology of T} by A10;
      the topology of space(Intersection Closed_Partitions T) =
    {A where A is Subset of
    (Intersection Closed_Partitions T) : union A in the topology of T}
              by BORSUK_1:def 10;
    then A12:  I in the topology of T_1-reflex T by A11,Def6;
    reconsider I as Subset of
    space(Intersection Closed_Partitions T)
              by BORSUK_1:def 10;
    reconsider I as Subset of T_1-reflex T by Def6;
      I = (the carrier of space(Intersection Closed_Partitions T)) \ {p}
              by BORSUK_1:def 10
     .= (the carrier of T_1-reflex T) \ {p} by Def6
     .= ([#] T_1-reflex T) \ {p} by PRE_TOPC:def 3;
    then ([#] T_1-reflex T) \ {p} is open by A12,PRE_TOPC:def 5;
    hence {p} is closed by PRE_TOPC:def 6;
   end;
   hence thesis by URYSOHN1:27;
  end;

definition let T;
 cluster T_1-reflex T -> being_T1;
 coherence by Th12;
end;

definition
 let T be non empty TopSpace;
  func T_1-reflect T -> continuous map of T,T_1-reflex T equals :Def7:
   Proj Intersection Closed_Partitions T;
  correctness
  proof
      space Intersection Closed_Partitions T = T_1-reflex T by Def6;
    hence thesis;
  end;
end;

theorem Th13:
 for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds
   T1 is being_T1 implies
   ({f"{z} where z is Element of T1 : z in rng f}
   is a_partition of the carrier of T) &
   (for A being Subset of T st
   A in {f"{z} where z is Element of T1 : z in rng f} holds
   A is closed)
    proof
     let T,T1 be non empty TopSpace;
     let f be continuous map of T,T1;
A1: dom f = the carrier of T & rng f c= the carrier of T1
      by FUNCT_2:def 1,RELSET_1:12;
     assume
A2: T1 is being_T1;
     thus {f"{z} where z is Element of T1 : z in rng f}
      is a_partition of the carrier of T
       proof
          {f"{z} where z is Element of T1 : z in rng f} c=
        bool the carrier of T
         proof
          let y;
          assume y in {f"{z} where z is Element of T1 : z in rng f};
          then consider z being Element of T1 such that
A3:        y = f"{z} & z in rng f;
          thus thesis by A3;
         end;
        then reconsider fz = {f"{z} where z is Element of T1 : z in rng f}
          as Subset-Family of T by SETFAM_1:def 7;
        reconsider fz as Subset-Family of T;
A4:     union fz = the carrier of T
         proof
          thus union fz c= the carrier of T;
          thus the carrier of T c= union fz
           proof
            let y;
            assume
A5:          y in the carrier of T;
            consider z being set such that
A6:          z = f.y;
A7:       z in rng f by A1,A5,A6,FUNCT_1:def 5;
            then reconsider z as Element of T1 by A1;
A8:          f"{z} in fz by A7;
              f.y in {f.y} by TARSKI:def 1;
            then y in f"{z} by A1,A5,A6,FUNCT_1:def 13;
            hence thesis by A8,TARSKI:def 4;
           end;
         end;
          for A being Subset of T st
             A in fz holds A <> {} &
             for B being Subset of T st
             B in fz holds A = B or A misses B
         proof
          let A be Subset of T;
          assume A in fz;
          then consider z being Element of T1 such that
A9:        A = f"{z} & z in rng f;
          consider y being set such that
A10:        y in dom f & z = f.y by A9,FUNCT_1:def 5;
            f.y in {f.y} by TARSKI:def 1;
          hence A <> {} by A9,A10,FUNCT_1:def 13;
          let B be Subset of T;
          assume B in fz;
          then consider w being Element of T1 such that
A11:        B = f"{w} & w in rng f;
          thus A = B or A misses B
           proof
              now
             assume not A misses B;
             then consider v being set such that
A12:           v in A & v in B by XBOOLE_0:3;
               f.v in {z} & f.v in {w} by A9,A11,A12,FUNCT_1:def 13;
             then f.v = z & f.v = w by TARSKI:def 1;
             hence A = B by A9,A11;
            end;
            hence thesis;
           end;
         end;
        hence thesis by A4,EQREL_1:def 6;
       end;
     thus for A being Subset of T st
      A in {f"{z} where z is Element of T1 : z in rng f} holds A is closed
       proof
        let A be Subset of T;
        assume
          A in {f"{z} where z is Element of T1 : z in rng f};
        then consider z being Element of T1 such that
A13:    A = f"{z} & z in rng f;
          {z} is closed by A2,URYSOHN1:27;
        hence thesis by A13,PRE_TOPC:def 12;
       end;
    end;

theorem Th14:
 for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds
   T1 is being_T1 implies for w for x being Element of T holds
    w = EqClass(x,(Intersection Closed_Partitions T)) implies w c= f"{f.x}
     proof
      let T,T1 be non empty TopSpace;
      let f be continuous map of T,T1;
A1: dom f = the carrier of T & rng f c= the carrier of T1
      by FUNCT_2:def 1,RELSET_1:12;
      assume
A2:  T1 is being_T1;
      let w be set;
      let x be Element of T;
      assume
A3:   w = EqClass(x,(Intersection Closed_Partitions T));
      let y be set;
      assume
A4:    y in w;
      reconsider fz = {f"{z} where z is Element of T1 : z in rng f}
       as a_partition of the carrier of T by A2,Th13;
        for A being Subset of T st A in fz holds A is closed by A2,Th13;
      then fz is closed by TOPS_2:def 2;
      then fz in {B where B is a_partition of the carrier of T : B is closed};
      then fz in Closed_Partitions T by Def5;
then A5:   EqClass(x,fz) in
      {EqClass(x,S) where S is a_partition of the carrier of T :
      S in Closed_Partitions T};
A6:  f"{f.x} = EqClass(x,fz)
       proof
          f.x in {f.x} by TARSKI:def 1;
        then A7:     x in f"{f.x} by A1,FUNCT_1:def 13;
A8:     f.x in rng f by A1,FUNCT_1:def 5;
        reconsider fx = f.x as Element of T1;
          f"{fx} in fz by A8;
        hence thesis by A7,Def1;
       end;
        EqClass(x,(Intersection Closed_Partitions T)) =
      meet{EqClass(x,S) where S is a_partition of the carrier of T :
      S in Closed_Partitions T} by Def4;
      hence y in f"{f.x} by A3,A4,A5,A6,SETFAM_1:def 1;
     end;

theorem Th15:
 for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds
   T1 is being_T1 implies
    for w st w in the carrier of T_1-reflex T ex z being Element of T1
    st z in rng f & w c= f"{z}
     proof
      let T,T1 be non empty TopSpace;
      let f be continuous map of T,T1;
      assume
A1:  T1 is being_T1;
      let w be set;
      assume
    w in the carrier of T_1-reflex T;
      then w in the carrier of space (Intersection Closed_Partitions T) by Def6
;
      then w in Intersection (Closed_Partitions T) by BORSUK_1:def 10;
      then consider x being Element of T such that
A2:    w = EqClass(x,Intersection (Closed_Partitions T)) by Th10;
      reconsider x as Element of T;
A3: dom f = the carrier of T & rng f c= the carrier of T1
      by FUNCT_2:def 1,RELSET_1:12;
      reconsider fx = f.x as Element of T1;
      take fx;
      thus thesis by A1,A2,A3,Th14,FUNCT_1:def 5;
     end;

:: The theorem on factorization

theorem Th16:
 for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds
   T1 is being_T1 implies
    ex h being continuous map of T_1-reflex T, T1 st
     f = h*T_1-reflect T
      proof
       let T,T1 be non empty TopSpace;
       let f be continuous map of T,T1;
       assume
A1:   T1 is being_T1;
       then reconsider fx = {f"{x} where x is Element of T1 : x in rng f}
        as a_partition of the carrier of T by Th13;
       set g = T_1-reflect T;
A2:  dom f = the carrier of T & rng f c= the carrier of T1
       by FUNCT_2:def 1,RELSET_1:12;
A3:  dom g = the carrier of T & rng g c= the carrier of T_1-reflex T
       by FUNCT_2:def 1,RELSET_1:12;
       defpred X[set,set] means for z being Element of T1 holds
       (z in rng f & $1 c= f"{z}) implies $2 = f"{z};
A4:    for y,w,v st y in the carrier of T_1-reflex T & X[y,w] & X[y,v]
       holds w = v
        proof
         let y,w,v;
         assume
A5:       y in the carrier of T_1-reflex T &
(for z being Element of T1 holds (z in rng f & y c= f"{z}) implies w = f"{z})
         &
(for z being Element of T1 holds (z in rng f & y c= f"{z}) implies v = f"{z});
         then y in the carrier of space Intersection(Closed_Partitions T) by
Def6;
         then y in Intersection(Closed_Partitions T) by BORSUK_1:def 10;
         then consider x being Element of T such that
A6:       y = EqClass(x,Intersection(Closed_Partitions T)) by Th10;
         reconsider x as Element of T;
         reconsider fix = f.x as Element of T1;
           fix in rng f & y c= f"{fix} by A1,A2,A6,Th14,FUNCT_1:def 5;
         then w = f"{fix} & v = f"{fix} by A5;
         hence w = v;
        end;

A7:    for y st y in the carrier of T_1-reflex T ex w st X[y,w]
        proof
         let y;
         assume y in the carrier of T_1-reflex T;
         then y in the carrier of space Intersection(Closed_Partitions T) by
Def6;
         then y in Intersection(Closed_Partitions T) by BORSUK_1:def 10;
         then consider x being Element of T such that
A8:       y = EqClass(x,Intersection(Closed_Partitions T)) by Th10;
         reconsider x as Element of T;
         set w = f"{f.x};
         take w;
         let z be Element of T1;
         assume
A9:      z in rng f & y c= f"{z};
         then A10:      f"{z} in fx;
A11:      f.x in rng f by A2,FUNCT_1:def 5;
         reconsider fix = f.x as Element of T1;
           f"{fix} in fx by A11;
         then A12:     w misses f"{z} or w = f"{z} by A10,EQREL_1:def 6;
A13:     y c= w & y c= f"{z} by A1,A8,A9,Th14;
           y is non empty by A8,Def1;
         then consider z1 being set such that
A14:      z1 in y by XBOOLE_0:def 1;
         thus w = f"{z} by A12,A13,A14,XBOOLE_0:3;
        end;

       consider h1 being Function such that
A15:     dom h1 = the carrier of T_1-reflex T &
       for y st y in the carrier of T_1-reflex T
           holds X[y,h1.y] from FuncEx(A4,A7);
        defpred X[set,set] means for z being Element of T1 holds
       (z in rng f & $1 = f"{z}) implies $2 = z;
A16:    for y,w,v st y in fx & X[y,w] & X[y,v]
       holds w = v
        proof
         let y,w,v;
         assume
A17: y in fx &
   (for z being Element of T1 holds (z in rng f & y = f"{z}) implies w = z)
    &
   (for z being Element of T1 holds (z in rng f & y = f"{z}) implies v = z);
         then consider z1 being Element of T1 such that
A18:      y = f"{z1} & z1 in rng f;
           w = z1 & v = z1 by A17,A18;
         hence thesis;
        end;

A19:    for y st y in fx ex w st X[y,w]
        proof
         let y be set;
         assume y in fx;
         then consider w being Element of T1 such that
A20:       y = f"{w} & w in rng f;
         take w;
         let z be Element of T1;
         assume
A21:     z in rng f & y = f"{z};
           now
          assume
A22:       z <> w;
          consider v being set such that
A23:        v in dom f & z = f.v by A21,FUNCT_1:def 5;
            z in {z} by TARSKI:def 1;
          then v in f"{w} by A20,A21,A23,FUNCT_1:def 13;
          then f.v in {w} by FUNCT_1:def 13;
          hence contradiction by A22,A23,TARSKI:def 1;
         end;
         hence w = z;
        end;
       consider h2 being Function such that
A24:     dom h2 = fx &
       for y st y in fx holds X[y,h2.y] from FuncEx(A16,A19);
       set h = h2*h1;
A25:  dom h = the carrier of T_1-reflex T
        proof
         thus dom h c= the carrier of T_1-reflex T by A15,RELAT_1:44;
         let z be set;
         assume
A26:     z in the carrier of T_1-reflex T;
         then consider w being Element of T1 such that
A27:       w in rng f & z c= f"{w} by A1,Th15;
           h1.z = f"{w} by A15,A26,A27;
         then h1.z in dom h2 by A24,A27;
         hence z in dom h by A15,A26,FUNCT_1:21;
        end;
A28:  rng h c= rng h2
        proof
         let z be set;
         thus thesis by FUNCT_1:25;
        end;
         rng h2 c= the carrier of T1
        proof
         let y;
         assume y in rng h2;
         then consider w being set such that
A29:       w in dom h2 & y = h2.w by FUNCT_1:def 5;
         consider x being Element of T1 such that
A30:       w = f"{x} & x in rng f by A24,A29;
           h2.w = x by A24,A29,A30;
         hence y in the carrier of T1 by A29;
        end;
       then A31:    rng h c= the carrier of T1 by A28,XBOOLE_1:1;

A32:  dom (h*g) = the carrier of T
        proof
         thus dom (h*g) c= the carrier of T by A3,RELAT_1:44;
         let y be set;
         assume
A33:       y in the carrier of T;
         then g.y in rng g by A3,FUNCT_1:def 5;
         hence y in dom (h*g) by A3,A25,A33,FUNCT_1:21;
        end;

A34:    f = h*g
        proof
           for x being set st x in dom f holds f.x = (h*g).x
          proof
           let x be set;
           assume
A35:         x in dom f;
           then g.x in rng g by A2,A3,FUNCT_1:def 5;
           then g.x in the carrier of T_1-reflex T by A3;
           then g.x in the carrier of space (Intersection Closed_Partitions T)
                           by Def6;
           then g.x in Intersection (Closed_Partitions T) by BORSUK_1:def 10;
           then consider y being Element of T such that
A36:         g.x = EqClass(y,Intersection (Closed_Partitions T)) by Th10;
           reconsider x as Element of T by A2,A35;
             g = Proj (Intersection Closed_Partitions T) by Def7;
           then g = proj (Intersection Closed_Partitions T) by BORSUK_1:def 11
;
           then A37:       x in g.x by BORSUK_1:def 1;
             x in EqClass(x,Intersection (Closed_Partitions T)) by Def1;
           then A38:       EqClass(x,Intersection (Closed_Partitions T)) meets
           EqClass(y,Intersection (Closed_Partitions T)) by A36,A37,XBOOLE_0:3;
           reconsider fix = f.x as Element of T1;
             g.x = EqClass(x,Intersection (Closed_Partitions T)) by A36,A38,Th9
;
           then A39:       fix in rng f & g.x c= f"{fix} by A1,A35,Th14,FUNCT_1
:def 5;
then A40:      f"{fix} in fx;
             (h*g).x = (h2*h1).(g.x) by A32,FUNCT_1:22
                  .= h2.(h1.(g.x)) by A15,FUNCT_1:23
                  .= h2.(f"{fix}) by A15,A39
                  .= f.x by A24,A39,A40;
           hence thesis;
          end;
        hence thesis by A2,A32,FUNCT_1:9;
       end;
       reconsider h as Function of
       the carrier of T_1-reflex T,the carrier of T1
                    by A25,A31,FUNCT_2:def 1,RELSET_1:11;
       reconsider h as map of T_1-reflex T,T1;
         h is continuous
        proof
         let y be Subset of T1;
         assume
A41:     y is closed;
         reconsider hy = h"y as Subset of
           space Intersection(Closed_Partitions T) by Def6;
           union hy c= the carrier of T
          proof
           let z1 be set;
           assume z1 in union hy;
           then consider z2 being set such that
A42:        z1 in z2 & z2 in hy by TARSKI:def 4;
             z2 in the carrier of space Intersection(Closed_Partitions T) by
A42
;
           then z2 in Intersection(Closed_Partitions T) by BORSUK_1:def 10;
           hence thesis by A42;
          end;
         then reconsider uhy = union hy as Subset of T;
A43:      T_1-reflex T = space (Intersection Closed_Partitions T) by Def6;
           (h*g)"y is closed by A34,A41,PRE_TOPC:def 12;
         then g"(h"y) is closed by RELAT_1:181;
         then (Proj (Intersection Closed_Partitions T))"hy is closed by Def7;
         then uhy is closed by Th2; hence h"y is closed by A43,Th5;
        end;
       then reconsider h as continuous map of T_1-reflex T,T1;
       take h;
       thus thesis by A34;
      end;

definition
 let T,S be non empty TopSpace;
 let f be continuous map of T,S;
  func T_1-reflex f -> continuous map of T_1-reflex T, T_1-reflex S means
   (T_1-reflect S) * f = it * (T_1-reflect T);
existence by Th16;
uniqueness
 proof
  let g1,g2 be continuous map of T_1-reflex T, T_1-reflex S;
  assume that
A1: (T_1-reflect S) * f = g1 * (T_1-reflect T) and
A2: (T_1-reflect S) * f = g2 * (T_1-reflect T);
A3: dom g1 = the carrier of (T_1-reflex T) &
   dom g2 = the carrier of (T_1-reflex T) by FUNCT_2:def 1;
    now
   let x be set;
   assume x in dom g1;
   then A4: x in the carrier of T_1-reflex T by FUNCT_2:def 1;
A5: the carrier of T_1-reflex T = the carrier of
                    space (Intersection Closed_Partitions T) by Def6
    .= (Intersection Closed_Partitions T) by BORSUK_1:def 10;
   then consider y being Element of T such that
A6: x = EqClass(y,(Intersection (Closed_Partitions T))) by A4,Th10;
 A7: dom (T_1-reflect T) = the carrier of T &
   rng (T_1-reflect T) c= the carrier of T_1-reflex T
        by FUNCT_2:def 1,RELSET_1:12;
   reconsider y as Element of T;
     T_1-reflect T = Proj (Intersection Closed_Partitions T) by Def7
                      .= proj (Intersection Closed_Partitions T)
                               by BORSUK_1:def 11;
   then A8: y in (T_1-reflect T).y & y in x by A6,Def1,BORSUK_1:def 1;
   set ty=(T_1-reflect T).y;
     ty in (Intersection Closed_Partitions T) by A5;
   then A9: ty misses x or ty = x by A4,A5,EQREL_1:def 6;
   hence g2.x = (g2 * (T_1-reflect T)).y by A7,A8,FUNCT_1:23,XBOOLE_0:3
            .= g1.x by A1,A2,A7,A8,A9,FUNCT_1:23,XBOOLE_0:3;
  end;
  hence g1=g2 by A3,FUNCT_1:9;
 end;
end;

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