Article T_0TOPSP, MML version 4.99.1005

:: T_0TOPSP:th 2
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of bool b1
      st for b5, b6 being Element of b1
              st b5 in b4 & b3 . b5 = b3 . b6
           holds b6 in b4
   holds b3 " (b3 .: b4) = b4;

:: T_0TOPSP:prednot 1 => T_0TOPSP:pred 1
definition
  let a1, a2 be TopStruct;
  pred A1,A2 are_homeomorphic means
    ex b1 being Function-like quasi_total Relation of the carrier of a1,the carrier of a2 st
       b1 is being_homeomorphism(a1, a2);
end;

:: T_0TOPSP:dfs 1
definiens
  let a1, a2 be TopStruct;
To prove
     a1,a2 are_homeomorphic
it is sufficient to prove
  thus ex b1 being Function-like quasi_total Relation of the carrier of a1,the carrier of a2 st
       b1 is being_homeomorphism(a1, a2);

:: T_0TOPSP:def 1
theorem
for b1, b2 being TopStruct holds
   b1,b2 are_homeomorphic
iff
   ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 st
      b3 is being_homeomorphism(b1, b2);

:: T_0TOPSP:attrnot 1 => T_0TOPSP:attr 1
definition
  let a1, a2 be TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is open means
    for b1 being Element of bool the carrier of a1
          st b1 is open(a1)
       holds a3 .: b1 is open(a2);
end;

:: T_0TOPSP:dfs 2
definiens
  let a1, a2 be TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is open
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 is open(a1)
       holds a3 .: b1 is open(a2);

:: T_0TOPSP:def 2
theorem
for b1, b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is open(b1, b2)
   iff
      for b4 being Element of bool the carrier of b1
            st b4 is open(b1)
         holds b3 .: b4 is open(b2);

:: T_0TOPSP:funcnot 1 => T_0TOPSP:func 1
definition
  let a1 be non empty TopStruct;
  func Indiscernibility A1 -> symmetric transitive total Relation of the carrier of a1,the carrier of a1 means
    for b1, b2 being Element of the carrier of a1 holds
       [b1,b2] in it
    iff
       for b3 being Element of bool the carrier of a1
             st b3 is open(a1)
          holds    b1 in b3
          iff
             b2 in b3;
end;

:: T_0TOPSP:def 3
theorem
for b1 being non empty TopStruct
for b2 being symmetric transitive total Relation of the carrier of b1,the carrier of b1 holds
      b2 = Indiscernibility b1
   iff
      for b3, b4 being Element of the carrier of b1 holds
         [b3,b4] in b2
      iff
         for b5 being Element of bool the carrier of b1
               st b5 is open(b1)
            holds    b3 in b5
            iff
               b4 in b5;

:: T_0TOPSP:funcnot 2 => T_0TOPSP:func 2
definition
  let a1 be non empty TopStruct;
  func Indiscernible A1 -> non empty a_partition of the carrier of a1 equals
    Class Indiscernibility a1;
end;

:: T_0TOPSP:def 4
theorem
for b1 being non empty TopStruct holds
   Indiscernible b1 = Class Indiscernibility b1;

:: T_0TOPSP:funcnot 3 => T_0TOPSP:func 3
definition
  let a1 be non empty TopSpace-like TopStruct;
  func T_0-reflex A1 -> TopSpace-like TopStruct equals
    space Indiscernible a1;
end;

:: T_0TOPSP:def 5
theorem
for b1 being non empty TopSpace-like TopStruct holds
   T_0-reflex b1 = space Indiscernible b1;

:: T_0TOPSP:funcreg 1
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster T_0-reflex a1 -> non empty TopSpace-like;
end;

:: T_0TOPSP:funcnot 4 => T_0TOPSP:func 4
definition
  let a1 be non empty TopSpace-like TopStruct;
  func T_0-canonical_map A1 -> Function-like quasi_total continuous Relation of the carrier of a1,the carrier of T_0-reflex a1 equals
    Proj Indiscernible a1;
end;

:: T_0TOPSP:def 6
theorem
for b1 being non empty TopSpace-like TopStruct holds
   T_0-canonical_map b1 = Proj Indiscernible b1;

:: T_0TOPSP:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of T_0-reflex b1 holds
      b2 is open(T_0-reflex b1)
   iff
      union b2 in the topology of b1;

:: T_0TOPSP:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being set holds
      b2 is Element of the carrier of T_0-reflex b1
   iff
      ex b3 being Element of the carrier of b1 st
         b2 = Class(Indiscernibility b1,b3);

:: T_0TOPSP:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   (T_0-canonical_map b1) . b2 = Class(Indiscernibility b1,b2);

:: T_0TOPSP:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1 holds
   (T_0-canonical_map b1) . b3 = (T_0-canonical_map b1) . b2
iff
   [b3,b2] in Indiscernibility b1;

:: T_0TOPSP:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
   st b2 is open(b1)
for b3, b4 being Element of the carrier of b1
      st b3 in b2 &
         (T_0-canonical_map b1) . b3 = (T_0-canonical_map b1) . b4
   holds b4 in b2;

:: T_0TOPSP:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
   st b2 is open(b1)
for b3 being Element of bool the carrier of b1
      st b3 in Indiscernible b1 & b3 meets b2
   holds b3 c= b2;

:: T_0TOPSP:th 12
theorem
for b1 being non empty TopSpace-like TopStruct holds
   T_0-canonical_map b1 is open(b1, T_0-reflex b1);

:: T_0TOPSP:attrnot 2 => T_0TOPSP:attr 2
definition
  let a1 be TopStruct;
  attr a1 is discerning means
    (a1 is not empty) implies for b1, b2 being Element of the carrier of a1
          st b1 <> b2
       holds ex b3 being Element of bool the carrier of a1 st
          b3 is open(a1) &
           (b1 in b3 & not b2 in b3 or b2 in b3 & not b1 in b3);
end;

:: T_0TOPSP:dfs 7
definiens
  let a1 be TopStruct;
To prove
     a1 is discerning
it is sufficient to prove
  thus (a1 is not empty) implies for b1, b2 being Element of the carrier of a1
          st b1 <> b2
       holds ex b3 being Element of bool the carrier of a1 st
          b3 is open(a1) &
           (b1 in b3 & not b2 in b3 or b2 in b3 & not b1 in b3);

:: T_0TOPSP:def 7
theorem
for b1 being TopStruct holds
      b1 is discerning
   iff
      (b1 is empty or for b2, b3 being Element of the carrier of b1
            st b2 <> b3
         holds ex b4 being Element of bool the carrier of b1 st
            b4 is open(b1) &
             (b2 in b4 & not b3 in b4 or b3 in b4 & not b2 in b4));

:: T_0TOPSP:exreg 1
registration
  cluster non empty TopSpace-like discerning TopStruct;
end;

:: T_0TOPSP:modenot 1
definition
  mode T_0-TopSpace is non empty TopSpace-like discerning TopStruct;
end;

:: T_0TOPSP:th 13
theorem
for b1 being non empty TopSpace-like TopStruct holds
   T_0-reflex b1 is non empty TopSpace-like discerning TopStruct;

:: T_0TOPSP:th 14
theorem
for b1, b2 being non empty TopSpace-like TopStruct
      st ex b3 being Function-like quasi_total Relation of the carrier of T_0-reflex b2,the carrier of T_0-reflex b1 st
           b3 is being_homeomorphism(T_0-reflex b2, T_0-reflex b1) & T_0-canonical_map b1,b3 * T_0-canonical_map b2 are_fiberwise_equipotent
   holds b1,b2 are_homeomorphic;

:: T_0TOPSP:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like discerning TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b4, b5 being Element of the carrier of b1
      st [b4,b5] in Indiscernibility b1
   holds b3 . b4 = b3 . b5;

:: T_0TOPSP:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like discerning TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
   b3 .: Class(Indiscernibility b1,b4) = {b3 . b4};

:: T_0TOPSP:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like discerning TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 holds
   ex b4 being Function-like quasi_total continuous Relation of the carrier of T_0-reflex b1,the carrier of b2 st
      b3 = b4 * T_0-canonical_map b1;