Article TOPS_3, MML version 4.99.1005

:: TOPS_3:th 1
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   (b2 = {} b1 implies b2 ` = [#] b1) & (b2 ` = [#] b1 implies b2 = {} b1) & (b2 = {} implies b2 ` = the carrier of b1) & (b2 ` = the carrier of b1 implies b2 = {});

:: TOPS_3:th 2
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   (b2 = [#] b1 implies b2 ` = {} b1) & (b2 ` = {} b1 implies b2 = [#] b1) & (b2 = the carrier of b1 implies b2 ` = {}) & (b2 ` = {} implies b2 = the carrier of b1);

:: TOPS_3:th 3
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Int b2) /\ Cl b3 c= Cl (b2 /\ b3);

:: TOPS_3:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Int (b2 \/ b3) c= (Cl b2) \/ Int b3;

:: TOPS_3:th 5
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b3 is closed(b1)
   holds Int (b3 \/ b2) c= b3 \/ Int b2;

:: TOPS_3:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b3 is closed(b1)
   holds Int (b3 \/ b2) = Int (b3 \/ Int b2);

:: TOPS_3:th 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 misses Int Cl b2
   holds Int Cl b2 = {};

:: TOPS_3:th 8
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 \/ Cl Int b2 = the carrier of b1
   holds Cl Int b2 = the carrier of b1;

:: TOPS_3:attrnot 1 => TOPS_1:attr 2
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is boundary means
    Int a2 = {};
end;

:: TOPS_3:dfs 1
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is boundary
it is sufficient to prove
  thus Int a2 = {};

:: TOPS_3:def 1
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is boundary(b1)
   iff
      Int b2 = {};

:: TOPS_3:th 9
theorem
for b1 being TopSpace-like TopStruct holds
   {} b1 is boundary(b1);

:: TOPS_3:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is boundary(b1)
   holds b2 <> the carrier of b1;

:: TOPS_3:th 11
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is boundary(b1) & b3 c= b2
   holds b3 is boundary(b1);

:: TOPS_3:th 12
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is boundary(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b2 ` c= b3 & b3 is closed(b1)
         holds b3 = the carrier of b1;

:: TOPS_3:th 13
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is boundary(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 <> {} & b3 is open(b1)
         holds b2 ` meets b3;

:: TOPS_3:th 14
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is boundary(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 is closed(b1)
         holds Int b3 = Int (b3 \/ b2);

:: TOPS_3:th 15
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is boundary(b1)
   holds b2 /\ b3 is boundary(b1);

:: TOPS_3:attrnot 2 => TOPS_1:attr 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is dense means
    Cl a2 = the carrier of a1;
end;

:: TOPS_3:dfs 2
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is dense
it is sufficient to prove
  thus Cl a2 = the carrier of a1;

:: TOPS_3:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is dense(b1)
   iff
      Cl b2 = the carrier of b1;

:: TOPS_3:th 16
theorem
for b1 being TopSpace-like TopStruct holds
   [#] b1 is dense(b1);

:: TOPS_3:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is dense(b1)
   holds b2 <> {};

:: TOPS_3:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is dense(b1)
   iff
      b2 ` is boundary(b1);

:: TOPS_3:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is dense(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b2 c= b3 & b3 is closed(b1)
         holds b3 = the carrier of b1;

:: TOPS_3:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is dense(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 is open(b1)
         holds Cl b3 = Cl (b3 /\ b2);

:: TOPS_3:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is dense(b1)
   holds b2 \/ b3 is dense(b1);

:: TOPS_3:attrnot 3 => TOPS_1:attr 3
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is nowhere_dense means
    Int Cl a2 = {};
end;

:: TOPS_3:dfs 3
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is nowhere_dense
it is sufficient to prove
  thus Int Cl a2 = {};

:: TOPS_3:def 3
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is nowhere_dense(b1)
   iff
      Int Cl b2 = {};

:: TOPS_3:th 22
theorem
for b1 being non empty TopSpace-like TopStruct holds
   {} b1 is nowhere_dense(b1);

:: TOPS_3:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is nowhere_dense(b1)
   holds b2 <> the carrier of b1;

:: TOPS_3:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is nowhere_dense(b1)
   holds Cl b2 is nowhere_dense(b1);

:: TOPS_3:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is nowhere_dense(b1)
   holds b2 is not dense(b1);

:: TOPS_3:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is nowhere_dense(b1) & b3 c= b2
   holds b3 is nowhere_dense(b1);

:: TOPS_3:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is nowhere_dense(b1)
   iff
      ex b3 being Element of bool the carrier of b1 st
         b2 c= b3 & b3 is closed(b1) & b3 is boundary(b1);

:: TOPS_3:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is nowhere_dense(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 <> {} & b3 is open(b1)
         holds ex b4 being Element of bool the carrier of b1 st
            b4 c= b3 & b4 <> {} & b4 is open(b1) & b2 misses b4;

:: TOPS_3:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is nowhere_dense(b1)
   holds b2 /\ b3 is nowhere_dense(b1);

:: TOPS_3:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is nowhere_dense(b1) & b3 is boundary(b1)
   holds b2 \/ b3 is boundary(b1);

:: TOPS_3:attrnot 4 => TOPS_3:attr 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is everywhere_dense means
    Cl Int a2 = [#] a1;
end;

:: TOPS_3:dfs 4
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is everywhere_dense
it is sufficient to prove
  thus Cl Int a2 = [#] a1;

:: TOPS_3:def 4
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is everywhere_dense(b1)
   iff
      Cl Int b2 = [#] b1;

:: TOPS_3:attrnot 5 => TOPS_3:attr 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is everywhere_dense means
    Cl Int a2 = the carrier of a1;
end;

:: TOPS_3:dfs 5
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is everywhere_dense
it is sufficient to prove
  thus Cl Int a2 = the carrier of a1;

:: TOPS_3:def 5
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is everywhere_dense(b1)
   iff
      Cl Int b2 = the carrier of b1;

:: TOPS_3:th 31
theorem
for b1 being non empty TopSpace-like TopStruct holds
   [#] b1 is everywhere_dense(b1);

:: TOPS_3:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1)
   holds Int b2 is everywhere_dense(b1);

:: TOPS_3:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1)
   holds b2 is dense(b1);

:: TOPS_3:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1)
   holds b2 <> {};

:: TOPS_3:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is everywhere_dense(b1)
   iff
      Int b2 is dense(b1);

:: TOPS_3:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is open(b1) & b2 is dense(b1)
   holds b2 is everywhere_dense(b1);

:: TOPS_3:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1)
   holds b2 is not boundary(b1);

:: TOPS_3:th 38
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1) & b2 c= b3
   holds b3 is everywhere_dense(b1);

:: TOPS_3:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is everywhere_dense(b1)
   iff
      b2 ` is nowhere_dense(b1);

:: TOPS_3:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is nowhere_dense(b1)
   iff
      b2 ` is everywhere_dense(b1);

:: TOPS_3:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is everywhere_dense(b1)
   iff
      ex b3 being Element of bool the carrier of b1 st
         b3 c= b2 & b3 is open(b1) & b3 is dense(b1);

:: TOPS_3:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is everywhere_dense(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 <> the carrier of b1 & b3 is closed(b1)
         holds ex b4 being Element of bool the carrier of b1 st
            b3 c= b4 & b4 <> the carrier of b1 & b4 is closed(b1) & b2 \/ b4 = the carrier of b1;

:: TOPS_3:th 43
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1)
   holds b2 \/ b3 is everywhere_dense(b1);

:: TOPS_3:th 44
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1) & b3 is everywhere_dense(b1)
   holds b2 /\ b3 is everywhere_dense(b1);

:: TOPS_3:th 45
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1) & b3 is dense(b1)
   holds b2 /\ b3 is dense(b1);

:: TOPS_3:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is dense(b1) & b3 is nowhere_dense(b1)
   holds b2 \ b3 is dense(b1);

:: TOPS_3:th 47
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1) & b3 is boundary(b1)
   holds b2 \ b3 is dense(b1);

:: TOPS_3:th 48
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1) & b3 is nowhere_dense(b1)
   holds b2 \ b3 is everywhere_dense(b1);

:: TOPS_3:th 49
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1)
   holds ex b3, b4 being Element of bool the carrier of b1 st
      b3 is open(b1) & b3 is dense(b1) & b4 is nowhere_dense(b1) & b3 \/ b4 = b2 & b3 misses b4;

:: TOPS_3:th 50
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is everywhere_dense(b1)
   holds ex b3, b4 being Element of bool the carrier of b1 st
      b3 is open(b1) & b3 is dense(b1) & b4 is closed(b1) & b4 is boundary(b1) & b3 \/ (b2 /\ b4) = b2 & b3 misses b4 & b3 \/ b4 = the carrier of b1;

:: TOPS_3:th 51
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is nowhere_dense(b1)
   holds ex b3, b4 being Element of bool the carrier of b1 st
      b3 is closed(b1) & b3 is boundary(b1) & b4 is everywhere_dense(b1) & b3 /\ b4 = b2 & b3 \/ b4 = the carrier of b1;

:: TOPS_3:th 52
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is nowhere_dense(b1)
   holds ex b3, b4 being Element of bool the carrier of b1 st
      b3 is closed(b1) & b3 is boundary(b1) & b4 is open(b1) & b4 is dense(b1) & b3 /\ (b2 \/ b4) = b2 & b3 misses b4 & b3 \/ b4 = the carrier of b1;

:: TOPS_3:th 53
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b4 c= b3
   holds Cl b4 c= Cl b3;

:: TOPS_3:th 54
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
      st b3 is closed(b1) & b3 c= the carrier of b2 & b4 c= b3 & b4 = b5
   holds Cl b4 = Cl b5;

:: TOPS_3:th 55
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty closed SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds Cl b3 = Cl b4;

:: TOPS_3:th 56
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 c= b4
   holds Int b3 c= Int b4;

:: TOPS_3:th 57
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
      st b3 is open(b1) & b3 c= the carrier of b2 & b4 c= b3 & b4 = b5
   holds Int b4 = Int b5;

:: TOPS_3:th 58
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty open SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds Int b3 = Int b4;

:: TOPS_3:th 59
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 c= b4 & b3 is dense(b1)
   holds b4 is dense(b2);

:: TOPS_3:th 60
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
      st b3 c= the carrier of b2 & b4 c= b3 & b4 = b5
   holds    b3 is dense(b1) & b5 is dense(b2)
   iff
      b4 is dense(b1);

:: TOPS_3:th 61
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 c= b4 & b3 is everywhere_dense(b1)
   holds b4 is everywhere_dense(b2);

:: TOPS_3:th 62
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
      st b3 is open(b1) & b3 c= the carrier of b2 & b4 c= b3 & b4 = b5
   holds    b3 is dense(b1) & b5 is everywhere_dense(b2)
   iff
      b4 is everywhere_dense(b1);

:: TOPS_3:th 63
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty open SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
      st b4 = the carrier of b2 & b3 = b5
   holds    b4 is dense(b1) & b5 is everywhere_dense(b2)
   iff
      b3 is everywhere_dense(b1);

:: TOPS_3:th 64
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
      st b3 c= the carrier of b2 & b4 c= b3 & b4 = b5
   holds    b3 is everywhere_dense(b1) & b5 is everywhere_dense(b2)
   iff
      b4 is everywhere_dense(b1);

:: TOPS_3:th 65
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 c= b4 & b4 is boundary(b2)
   holds b3 is boundary(b1);

:: TOPS_3:th 66
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
      st b3 is open(b1) & b3 c= the carrier of b2 & b4 c= b3 & b4 = b5 & b4 is boundary(b1)
   holds b5 is boundary(b2);

:: TOPS_3:th 67
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty open SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds    b3 is boundary(b1)
   iff
      b4 is boundary(b2);

:: TOPS_3:th 68
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 c= b4 & b4 is nowhere_dense(b2)
   holds b3 is nowhere_dense(b1);

:: TOPS_3:th 69
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
      st b3 is open(b1) & b3 c= the carrier of b2 & b4 c= b3 & b4 = b5 & b4 is nowhere_dense(b1)
   holds b5 is nowhere_dense(b2);

:: TOPS_3:th 70
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty open SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds    b3 is nowhere_dense(b1)
   iff
      b4 is nowhere_dense(b2);

:: TOPS_3:th 71
theorem
for b1, b2 being 1-sorted
   st the carrier of b1 = the carrier of b2
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2 holds
      b3 = b4
   iff
      b3 ` = b4 `;

:: TOPS_3:th 72
theorem
for b1, b2 being TopStruct
      st the carrier of b1 = the carrier of b2 &
         (for b3 being Element of bool the carrier of b1
         for b4 being Element of bool the carrier of b2
               st b3 = b4
            holds    b3 is open(b1)
            iff
               b4 is open(b2))
   holds TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#);

:: TOPS_3:th 73
theorem
for b1, b2 being TopStruct
      st the carrier of b1 = the carrier of b2 &
         (for b3 being Element of bool the carrier of b1
         for b4 being Element of bool the carrier of b2
               st b3 = b4
            holds    b3 is closed(b1)
            iff
               b4 is closed(b2))
   holds TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#);

:: TOPS_3:th 74
theorem
for b1, b2 being TopSpace-like TopStruct
      st the carrier of b1 = the carrier of b2 &
         (for b3 being Element of bool the carrier of b1
         for b4 being Element of bool the carrier of b2
               st b3 = b4
            holds Int b3 = Int b4)
   holds TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#);

:: TOPS_3:th 75
theorem
for b1, b2 being TopSpace-like TopStruct
      st the carrier of b1 = the carrier of b2 &
         (for b3 being Element of bool the carrier of b1
         for b4 being Element of bool the carrier of b2
               st b3 = b4
            holds Cl b3 = Cl b4)
   holds TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#);

:: TOPS_3:th 76
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b3 is open(b1)
   holds b4 is open(b2);

:: TOPS_3:th 77
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#)
   holds Int b3 = Int b4;

:: TOPS_3:th 78
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 c= b4 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#)
   holds Int b3 c= Int b4;

:: TOPS_3:th 79
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b3 is closed(b1)
   holds b4 is closed(b2);

:: TOPS_3:th 80
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#)
   holds Cl b3 = Cl b4;

:: TOPS_3:th 81
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 c= b4 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#)
   holds Cl b3 c= Cl b4;

:: TOPS_3:th 82
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b4 c= b3 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b3 is boundary(b1)
   holds b4 is boundary(b2);

:: TOPS_3:th 83
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 c= b4 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b3 is dense(b1)
   holds b4 is dense(b2);

:: TOPS_3:th 84
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b4 c= b3 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b3 is nowhere_dense(b1)
   holds b4 is nowhere_dense(b2);

:: TOPS_3:th 85
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 c= b4 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b3 is everywhere_dense(b1)
   holds b4 is everywhere_dense(b2);