Article T_1TOPSP, MML version 4.99.1005

:: T_1TOPSP:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1
for b3 being Element of bool the carrier of space b2 holds
   (Proj b2) " b3 = union b3;

:: T_1TOPSP:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1
for b3 being Element of bool the carrier of space b2
for b4 being Element of bool the carrier of b1
      st b4 = union b3
   holds    b3 is closed(space b2)
   iff
      b4 is closed(b1);

:: T_1TOPSP:th 11
theorem
for b1 being non empty TopSpace-like TopStruct holds
   {b2 where b2 is a_partition of the carrier of b1: b2 is closed(b1)} is partition-membered Family-Class of the carrier of b1;

:: T_1TOPSP:funcnot 1 => T_1TOPSP:func 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  func Closed_Partitions A1 -> non empty partition-membered Family-Class of the carrier of a1 equals
    {b1 where b1 is a_partition of the carrier of a1: b1 is closed(a1)};
end;

:: T_1TOPSP:def 5
theorem
for b1 being non empty TopSpace-like TopStruct holds
   Closed_Partitions b1 = {b2 where b2 is a_partition of the carrier of b1: b2 is closed(b1)};

:: T_1TOPSP:funcnot 2 => T_1TOPSP:func 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  func T_1-reflex A1 -> TopSpace-like TopStruct equals
    space Intersection Closed_Partitions a1;
end;

:: T_1TOPSP:def 6
theorem
for b1 being non empty TopSpace-like TopStruct holds
   T_1-reflex b1 = space Intersection Closed_Partitions b1;

:: T_1TOPSP:funcreg 1
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster T_1-reflex a1 -> non empty strict TopSpace-like;
end;

:: T_1TOPSP:th 12
theorem
for b1 being non empty TopSpace-like TopStruct holds
   T_1-reflex b1 is being_T1;

:: T_1TOPSP:funcreg 2
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster T_1-reflex a1 -> TopSpace-like being_T1;
end;

:: T_1TOPSP:exreg 1
registration
  cluster non empty TopSpace-like being_T1 TopStruct;
end;

:: T_1TOPSP:funcnot 3 => T_1TOPSP:func 3
definition
  let a1 be non empty TopSpace-like TopStruct;
  func T_1-reflect A1 -> Function-like quasi_total continuous Relation of the carrier of a1,the carrier of T_1-reflex a1 equals
    Proj Intersection Closed_Partitions a1;
end;

:: T_1TOPSP:def 7
theorem
for b1 being non empty TopSpace-like TopStruct holds
   T_1-reflect b1 = Proj Intersection Closed_Partitions b1;

:: T_1TOPSP:th 13
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
      st b2 is being_T1
   holds {b3 " {b4} where b4 is Element of the carrier of b2: b4 in rng b3} is a_partition of the carrier of b1 &
    (for b4 being Element of bool the carrier of b1
          st b4 in {b3 " {b5} where b5 is Element of the carrier of b2: b5 in rng b3}
       holds b4 is closed(b1));

:: T_1TOPSP:th 14
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
   st b2 is being_T1
for b4 being set
for b5 being Element of the carrier of b1
      st b4 = EqClass(b5,Intersection Closed_Partitions b1)
   holds b4 c= b3 " {b3 . b5};

:: T_1TOPSP:th 15
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
   st b2 is being_T1
for b4 being set
      st b4 in the carrier of T_1-reflex b1
   holds ex b5 being Element of the carrier of b2 st
      b5 in rng b3 & b4 c= b3 " {b5};

:: T_1TOPSP:th 16
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
      st b2 is being_T1
   holds ex b4 being Function-like quasi_total continuous Relation of the carrier of T_1-reflex b1,the carrier of b2 st
      b3 = b4 * T_1-reflect b1;

:: T_1TOPSP:funcnot 4 => T_1TOPSP:func 4
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2;
  func T_1-reflex A3 -> Function-like quasi_total continuous Relation of the carrier of T_1-reflex a1,the carrier of T_1-reflex a2 means
    (T_1-reflect a2) * a3 = it * T_1-reflect a1;
end;

:: T_1TOPSP:def 8
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total continuous Relation of the carrier of T_1-reflex b1,the carrier of T_1-reflex b2 holds
      b4 = T_1-reflex b3
   iff
      (T_1-reflect b2) * b3 = b4 * T_1-reflect b1;