Article TOPS_2, MML version 4.99.1005

:: TOPS_2:th 1
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1 holds
   b2 c= bool [#] b1;

:: TOPS_2:th 3
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1
for b3 being set
      st b3 c= b2
   holds b3 is Element of bool bool the carrier of b1;

:: TOPS_2:th 5
theorem
for b1 being non empty 1-sorted
for b2 being Element of bool bool the carrier of b1
      st b2 is_a_cover_of b1
   holds b2 <> {};

:: TOPS_2:th 6
theorem
for b1 being 1-sorted
for b2, b3 being Element of bool bool the carrier of b1 holds
(union b2) \ union b3 c= union (b2 \ b3);

:: TOPS_2:th 10
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
      b2 <> {}
   iff
      COMPLEMENT b2 <> {};

:: TOPS_2:th 11
theorem
for b1 being set
for b2 being Element of bool bool b1
      st b2 <> {}
   holds meet COMPLEMENT b2 = (union b2) `;

:: TOPS_2:th 12
theorem
for b1 being set
for b2 being Element of bool bool b1
      st b2 <> {}
   holds union COMPLEMENT b2 = (meet b2) `;

:: TOPS_2:th 13
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1 holds
      COMPLEMENT b2 is finite
   iff
      b2 is finite;

:: TOPS_2:attrnot 1 => TOPS_2:attr 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is open means
    for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is open(a1);
end;

:: TOPS_2:dfs 1
definiens
  let a1 be TopStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is open
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is open(a1);

:: TOPS_2:def 1
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is open(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 in b2
         holds b3 is open(b1);

:: TOPS_2:attrnot 2 => TOPS_2:attr 2
definition
  let a1 be TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is closed means
    for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is closed(a1);
end;

:: TOPS_2:dfs 2
definiens
  let a1 be TopStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is closed
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is closed(a1);

:: TOPS_2:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is closed(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 in b2
         holds b3 is closed(b1);

:: TOPS_2:th 16
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is closed(b1)
   iff
      COMPLEMENT b2 is open(b1);

:: TOPS_2:th 17
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is open(b1)
   iff
      COMPLEMENT b2 is closed(b1);

:: TOPS_2:th 18
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 c= b3 & b3 is open(b1)
   holds b2 is open(b1);

:: TOPS_2:th 19
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 c= b3 & b3 is closed(b1)
   holds b2 is closed(b1);

:: TOPS_2:th 20
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 is open(b1) & b3 is open(b1)
   holds b2 \/ b3 is open(b1);

:: TOPS_2:th 21
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 is open(b1)
   holds b2 /\ b3 is open(b1);

:: TOPS_2:th 22
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 is open(b1)
   holds b2 \ b3 is open(b1);

:: TOPS_2:th 23
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 is closed(b1) & b3 is closed(b1)
   holds b2 \/ b3 is closed(b1);

:: TOPS_2:th 24
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 is closed(b1)
   holds b2 /\ b3 is closed(b1);

:: TOPS_2:th 25
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 is closed(b1)
   holds b2 \ b3 is closed(b1);

:: TOPS_2:th 26
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is open(b1)
   holds union b2 is open(b1);

:: TOPS_2:th 27
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is open(b1) & b2 is finite
   holds meet b2 is open(b1);

:: TOPS_2:th 28
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is closed(b1) & b2 is finite
   holds union b2 is closed(b1);

:: TOPS_2:th 29
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is closed(b1)
   holds meet b2 is closed(b1);

:: TOPS_2:th 31
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool bool the carrier of b2 holds
   b3 is Element of bool bool the carrier of b1;

:: TOPS_2:th 32
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b2 holds
      b3 is open(b2)
   iff
      ex b4 being Element of bool the carrier of b1 st
         b4 is open(b1) & b4 /\ [#] b2 = b3;

:: TOPS_2:th 33
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being SubSpace of b1
   st b2 is open(b1)
for b4 being Element of bool the carrier of b3
      st b4 = b2
   holds b4 is open(b3);

:: TOPS_2:th 34
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being SubSpace of b1
   st b2 is closed(b1)
for b4 being Element of bool the carrier of b3
      st b4 = b2
   holds b4 is closed(b3);

:: TOPS_2:th 35
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being SubSpace of b1
   st b2 is open(b1)
for b4 being Element of bool bool the carrier of b3
      st b4 = b2
   holds b4 is open(b3);

:: TOPS_2:th 36
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being SubSpace of b1
   st b2 is closed(b1)
for b4 being Element of bool bool the carrier of b3
      st b4 = b2
   holds b4 is closed(b3);

:: TOPS_2:th 38
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b2 /\ b3 is Element of bool the carrier of b1 | b3;

:: TOPS_2:funcnot 1 => TOPS_2:func 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be Element of bool bool the carrier of a1;
  func A3 | A2 -> Element of bool bool the carrier of a1 | a2 means
    for b1 being Element of bool the carrier of a1 | a2 holds
          b1 in it
       iff
          ex b2 being Element of bool the carrier of a1 st
             b2 in a3 & b2 /\ a2 = b1;
end;

:: TOPS_2:def 3
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
for b4 being Element of bool bool the carrier of b1 | b2 holds
      b4 = b3 | b2
   iff
      for b5 being Element of bool the carrier of b1 | b2 holds
            b5 in b4
         iff
            ex b6 being Element of bool the carrier of b1 st
               b6 in b3 & b6 /\ b2 = b5;

:: TOPS_2:th 40
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of bool bool the carrier of b1
      st b3 c= b4
   holds b3 | b2 c= b4 | b2;

:: TOPS_2:th 41
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of bool bool the carrier of b1
      st b2 in b4
   holds b2 /\ b3 in b4 | b3;

:: TOPS_2:th 42
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of bool bool the carrier of b1
      st b2 c= union b4
   holds b2 /\ b3 c= union (b4 | b3);

:: TOPS_2:th 43
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
      st b2 c= union b3
   holds b2 = union (b3 | b2);

:: TOPS_2:th 44
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1 holds
   union (b3 | b2) c= union b3;

:: TOPS_2:th 45
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
      st b2 c= union (b3 | b2)
   holds b2 c= union b3;

:: TOPS_2:th 46
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
      st b3 is finite
   holds b3 | b2 is finite;

:: TOPS_2:th 47
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
      st b3 is open(b1)
   holds b3 | b2 is open(b1 | b2);

:: TOPS_2:th 48
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
      st b3 is closed(b1)
   holds b3 | b2 is closed(b1 | b2);

:: TOPS_2:th 49
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool bool the carrier of b2
      st b3 is open(b2)
   holds ex b4 being Element of bool bool the carrier of b1 st
      b4 is open(b1) &
       (for b5 being Element of bool the carrier of b1
             st b5 = [#] b2
          holds b3 = b4 | b5);

:: TOPS_2:th 50
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1 holds
   ex b4 being Relation-like Function-like set st
      proj1 b4 = b3 &
       proj2 b4 = b3 | b2 &
       (for b5 being set
          st b5 in b3
       for b6 being Element of bool the carrier of b1
             st b6 = b5
          holds b4 . b5 = b6 /\ b2);

:: TOPS_2:th 52
theorem
for b1, b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st ([#] b2 = {} implies [#] b1 = {})
   holds b3 " [#] b2 = [#] b1;

:: TOPS_2:th 54
theorem
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool bool the carrier of b2 holds
   (" b3) .: b4 is Element of bool bool the carrier of b1;

:: TOPS_2:th 55
theorem
for b1, b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st ([#] b2 = {} implies [#] b1 = {})
   holds    b3 is continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 is open(b1);

:: TOPS_2:th 56
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2 holds
         Cl (b3 " b4) c= b3 " Cl b4;

:: TOPS_2:th 57
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b1 holds
         b3 .: Cl b4 c= Cl (b3 .: b4);

:: TOPS_2:th 58
theorem
for b1, b2 being TopStruct
for b3 being non empty TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
      st b4 is continuous(b1, b3) & b5 is continuous(b3, b2)
   holds b5 * b4 is continuous(b1, b2);

:: TOPS_2:th 59
theorem
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool bool the carrier of b2
   st b3 is continuous(b1, b2) & b4 is open(b2)
for b5 being Element of bool bool the carrier of b1
      st b5 = (" b3) .: b4
   holds b5 is open(b1);

:: TOPS_2:th 60
theorem
for b1, b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool bool the carrier of b2
   st b3 is continuous(b1, b2) & b4 is closed(b2)
for b5 being Element of bool bool the carrier of b1
      st b5 = (" b3) .: b4
   holds b5 is closed(b1);

:: TOPS_2:funcnot 2 => TOPS_2:func 2
definition
  let a1, a2 be set;
  let a3 be Function-like quasi_total Relation of a1,a2;
  assume rng a3 = a2 & a3 is one-to-one;
  func A3 /" -> Function-like quasi_total Relation of a2,a1 equals
    a3 ";
end;

:: TOPS_2:def 4
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of b1,b2
      st rng b3 = b2 & b3 is one-to-one
   holds b3 /" = b3 ";

:: TOPS_2:funcnot 3 => TOPS_2:func 2
notation
  let a1, a2 be set;
  let a3 be Function-like quasi_total Relation of a1,a2;
  synonym a3 " for a3 /";
end;

:: TOPS_2:th 62
theorem
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st rng b3 = [#] b2 & b3 is one-to-one
   holds dom (b3 /") = [#] b2 & rng (b3 /") = [#] b1;

:: TOPS_2:th 63
theorem
for b1, b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st rng b3 = [#] b2 & b3 is one-to-one
   holds b3 /" is one-to-one;

:: TOPS_2:th 64
theorem
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st rng b3 = [#] b2 & b3 is one-to-one
   holds b3 /" /" = b3;

:: TOPS_2:th 65
theorem
for b1, b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st rng b3 = [#] b2 & b3 is one-to-one
   holds b3 /" * b3 = id dom b3 & b3 * (b3 /") = id rng b3;

:: TOPS_2:th 66
theorem
for b1 being 1-sorted
for b2, b3 being non empty 1-sorted
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
      st dom b4 = [#] b1 & rng b4 = [#] b2 & b4 is one-to-one & dom b5 = [#] b2 & rng b5 = [#] b3 & b5 is one-to-one
   holds (b5 * b4) /" = b4 /" * (b5 /");

:: TOPS_2:th 67
theorem
for b1, b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
      st rng b3 = [#] b2 & b3 is one-to-one
   holds b3 .: b4 = b3 /" " b4;

:: TOPS_2:th 68
theorem
for b1, b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b2
      st rng b3 = [#] b2 & b3 is one-to-one
   holds b3 " b4 = b3 /" .: b4;

:: TOPS_2:attrnot 3 => TOPS_2:attr 3
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 being_homeomorphism means
    dom a3 = [#] a1 & rng a3 = [#] a2 & a3 is one-to-one & a3 is continuous(a1, a2) & a3 /" is continuous(a2, a1);
end;

:: TOPS_2:dfs 5
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 being_homeomorphism
it is sufficient to prove
  thus dom a3 = [#] a1 & rng a3 = [#] a2 & a3 is one-to-one & a3 is continuous(a1, a2) & a3 /" is continuous(a2, a1);

:: TOPS_2:def 5
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 being_homeomorphism(b1, b2)
   iff
      dom b3 = [#] b1 & rng b3 = [#] b2 & b3 is one-to-one & b3 is continuous(b1, b2) & b3 /" is continuous(b2, b1);

:: TOPS_2:prednot 1 => TOPS_2:attr 3
notation
  let a1, a2 be TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  synonym a3 is_homeomorphism for being_homeomorphism;
end;

:: TOPS_2:th 70
theorem
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is being_homeomorphism(b1, b2)
   holds b3 /" is being_homeomorphism(b2, b1);

:: TOPS_2:th 71
theorem
for b1, b2, b3 being non empty TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
      st b4 is being_homeomorphism(b1, b2) & b5 is being_homeomorphism(b2, b3)
   holds b5 * b4 is being_homeomorphism(b1, b3);

:: TOPS_2:th 72
theorem
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_homeomorphism(b1, b2)
   iff
      dom b3 = [#] b1 &
       rng b3 = [#] b2 &
       b3 is one-to-one &
       (for b4 being Element of bool the carrier of b1 holds
             b4 is closed(b1)
          iff
             b3 .: b4 is closed(b2));

:: TOPS_2:th 73
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_homeomorphism(b1, b2)
   iff
      dom b3 = [#] b1 &
       rng b3 = [#] b2 &
       b3 is one-to-one &
       (for b4 being Element of bool the carrier of b2 holds
          b3 " Cl b4 = Cl (b3 " b4));

:: TOPS_2:th 74
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_homeomorphism(b1, b2)
   iff
      dom b3 = [#] b1 &
       rng b3 = [#] b2 &
       b3 is one-to-one &
       (for b4 being Element of bool the carrier of b1 holds
          b3 .: Cl b4 = Cl (b3 .: b4));

:: TOPS_2:th 75
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
      st b3 is continuous(b1, b2) & b4 is connected(b1)
   holds b3 .: b4 is connected(b2);

:: TOPS_2:th 76
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b2
      st b3 is being_homeomorphism(b1, b2) & b4 is connected(b2)
   holds b3 " b4 is connected(b1);