Article TOPGEN_1, MML version 4.99.1005

:: TOPGEN_1:th 1
theorem
for b1 being 1-sorted
for b2, b3 being Element of bool the carrier of b1 holds
   b2 meets b3 `
iff
   b2 \ b3 <> {};

:: TOPGEN_1:th 3
theorem
for b1 being 1-sorted holds
      b1 is countable
   iff
      Card [#] b1 c= alef 0;

:: TOPGEN_1:funcreg 1
registration
  let a1 be finite 1-sorted;
  cluster [#] a1 -> finite;
end;

:: TOPGEN_1:condreg 1
registration
  cluster finite -> countable (1-sorted);
end;

:: TOPGEN_1:exreg 1
registration
  cluster non empty countable 1-sorted;
end;

:: TOPGEN_1:exreg 2
registration
  cluster non empty TopSpace-like countable TopStruct;
end;

:: TOPGEN_1:funcreg 2
registration
  let a1 be countable 1-sorted;
  cluster [#] a1 -> countable;
end;

:: TOPGEN_1:exreg 3
registration
  cluster non empty TopSpace-like being_T1 TopStruct;
end;

:: TOPGEN_1:th 5
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Int b2 = (Cl (b2 `)) `;

:: TOPGEN_1:funcnot 1 => TOPGEN_1:func 1
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  func Fr A2 -> Element of bool bool the carrier of a1 means
    for b1 being Element of bool the carrier of a1 holds
          b1 in it
       iff
          ex b2 being Element of bool the carrier of a1 st
             b1 = Fr b2 & b2 in a2;
end;

:: TOPGEN_1:def 1
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
   b3 = Fr b2
iff
   for b4 being Element of bool the carrier of b1 holds
         b4 in b3
      iff
         ex b5 being Element of bool the carrier of b1 st
            b4 = Fr b5 & b5 in b2;

:: TOPGEN_1:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 = {}
   holds Fr b2 = {};

:: TOPGEN_1:th 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being Element of bool the carrier of b1
      st b2 = {b3}
   holds Fr b2 = {Fr b3};

:: TOPGEN_1:th 8
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 c= b3
   holds Fr b2 c= Fr b3;

:: TOPGEN_1:th 9
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
Fr (b2 \/ b3) = (Fr b2) \/ Fr b3;

:: TOPGEN_1:th 10
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   Fr b2 = (Cl b2) \ Int b2;

:: TOPGEN_1:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Fr b2
   iff
      for b4 being Element of bool the carrier of b1
            st b4 is open(b1) & b3 in b4
         holds b2 meets b4 & b4 \ b2 <> {};

:: TOPGEN_1:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Fr b2
   iff
      for b4 being Basis of b3
      for b5 being Element of bool the carrier of b1
            st b5 in b4
         holds b2 meets b5 & b5 \ b2 <> {};

:: TOPGEN_1:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Fr b2
   iff
      ex b4 being Basis of b3 st
         for b5 being Element of bool the carrier of b1
               st b5 in b4
            holds b2 meets b5 & b5 \ b2 <> {};

:: TOPGEN_1:th 14
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Fr (b2 /\ b3) c= ((Cl b2) /\ Fr b3) \/ ((Fr b2) /\ Cl b3);

:: TOPGEN_1:th 15
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   the carrier of b1 = ((Int b2) \/ Fr b2) \/ Int (b2 `);

:: TOPGEN_1:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is open(b1) & b2 is closed(b1)
   iff
      Fr b2 = {};

:: TOPGEN_1:prednot 1 => TOPGEN_1:pred 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be set;
  pred A3 is_an_accumulation_point_of A2 means
    a3 in Cl (a2 \ {a3});
end;

:: TOPGEN_1:dfs 2
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be set;
To prove
     a3 is_an_accumulation_point_of a2
it is sufficient to prove
  thus a3 in Cl (a2 \ {a3});

:: TOPGEN_1:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being set holds
      b3 is_an_accumulation_point_of b2
   iff
      b3 in Cl (b2 \ {b3});

:: TOPGEN_1:th 17
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being set
      st b3 is_an_accumulation_point_of b2
   holds b3 is Element of the carrier of b1;

:: TOPGEN_1:funcnot 2 => TOPGEN_1:func 2
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  func Der A2 -> Element of bool the carrier of a1 means
    for b1 being set
          st b1 in the carrier of a1
       holds    b1 in it
       iff
          b1 is_an_accumulation_point_of a2;
end;

:: TOPGEN_1:def 3
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b3 = Der b2
iff
   for b4 being set
         st b4 in the carrier of b1
      holds    b4 in b3
      iff
         b4 is_an_accumulation_point_of b2;

:: TOPGEN_1:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being set holds
      b3 in Der b2
   iff
      b3 is_an_accumulation_point_of b2;

:: TOPGEN_1:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Der b2
   iff
      for b4 being open Element of bool the carrier of b1
            st b3 in b4
         holds ex b5 being Element of the carrier of b1 st
            b5 in b2 /\ b4 & b3 <> b5;

:: TOPGEN_1:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Der b2
   iff
      for b4 being Basis of b3
      for b5 being Element of bool the carrier of b1
            st b5 in b4
         holds ex b6 being Element of the carrier of b1 st
            b6 in b2 /\ b5 & b3 <> b6;

:: TOPGEN_1:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Der b2
   iff
      ex b4 being Basis of b3 st
         for b5 being Element of bool the carrier of b1
               st b5 in b4
            holds ex b6 being Element of the carrier of b1 st
               b6 in b2 /\ b5 & b3 <> b6;

:: TOPGEN_1:prednot 2 => TOPGEN_1:pred 2
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be set;
  pred A3 is_isolated_in A2 means
    a3 in a2 & not a3 is_an_accumulation_point_of a2;
end;

:: TOPGEN_1:dfs 4
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be set;
To prove
     a3 is_isolated_in a2
it is sufficient to prove
  thus a3 in a2 & not a3 is_an_accumulation_point_of a2;

:: TOPGEN_1:def 4
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being set holds
      b3 is_isolated_in b2
   iff
      b3 in b2 & not b3 is_an_accumulation_point_of b2;

:: TOPGEN_1:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being set holds
      b3 in b2 \ Der b2
   iff
      b3 is_isolated_in b2;

:: TOPGEN_1:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 is_an_accumulation_point_of b2
   iff
      for b4 being open Element of bool the carrier of b1
            st b3 in b4
         holds ex b5 being Element of the carrier of b1 st
            b5 <> b3 & b5 in b2 & b5 in b4;

:: TOPGEN_1:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 is_isolated_in b2
   iff
      ex b4 being open Element of bool the carrier of b1 st
         b4 /\ b2 = {b3};

:: TOPGEN_1:attrnot 1 => TOPGEN_1:attr 1
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
  attr a2 is isolated means
    a2 is_isolated_in [#] a1;
end;

:: TOPGEN_1:dfs 5
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
To prove
     a2 is isolated
it is sufficient to prove
  thus a2 is_isolated_in [#] a1;

:: TOPGEN_1:def 5
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
      b2 is isolated(b1)
   iff
      b2 is_isolated_in [#] b1;

:: TOPGEN_1:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
      b2 is isolated(b1)
   iff
      {b2} is open(b1);

:: TOPGEN_1:funcnot 3 => TOPGEN_1:func 3
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  func Der A2 -> Element of bool bool the carrier of a1 means
    for b1 being Element of bool the carrier of a1 holds
          b1 in it
       iff
          ex b2 being Element of bool the carrier of a1 st
             b1 = Der b2 & b2 in a2;
end;

:: TOPGEN_1:def 6
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
   b3 = Der b2
iff
   for b4 being Element of bool the carrier of b1 holds
         b4 in b3
      iff
         ex b5 being Element of bool the carrier of b1 st
            b4 = Der b5 & b5 in b2;

:: TOPGEN_1:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 = {}
   holds Der b2 = {};

:: TOPGEN_1:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
      st b3 = {b2}
   holds Der b3 = {Der b2};

:: TOPGEN_1:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 c= b3
   holds Der b2 c= Der b3;

:: TOPGEN_1:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
Der (b2 \/ b3) = (Der b2) \/ Der b3;

:: TOPGEN_1:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Der b2 c= Cl b2;

:: TOPGEN_1:th 31
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Cl b2 = b2 \/ Der b2;

:: TOPGEN_1:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 c= b3
   holds Der b2 c= Der b3;

:: TOPGEN_1:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Der (b2 \/ b3) = (Der b2) \/ Der b3;

:: TOPGEN_1:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b1 is being_T1
   holds Der Der b2 c= Der b2;

:: TOPGEN_1:th 35
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b1 is being_T1
   holds Cl Der b2 = Der b2;

:: TOPGEN_1:funcreg 3
registration
  let a1 be non empty TopSpace-like being_T1 TopStruct;
  let a2 be Element of bool the carrier of a1;
  cluster Der a2 -> closed;
end;

:: TOPGEN_1:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
   union Der b2 c= Der union b2;

:: TOPGEN_1:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being set
      st b2 c= b3 & b4 is_an_accumulation_point_of b2
   holds b4 is_an_accumulation_point_of b3;

:: TOPGEN_1:attrnot 2 => TOPGEN_1:attr 2
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is dense-in-itself means
    a2 c= Der a2;
end;

:: TOPGEN_1:dfs 7
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is dense-in-itself
it is sufficient to prove
  thus a2 c= Der a2;

:: TOPGEN_1:def 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is dense-in-itself(b1)
   iff
      b2 c= Der b2;

:: TOPGEN_1:attrnot 3 => TOPGEN_1:attr 3
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is dense-in-itself means
    [#] a1 is dense-in-itself(a1);
end;

:: TOPGEN_1:dfs 8
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is dense-in-itself
it is sufficient to prove
  thus [#] a1 is dense-in-itself(a1);

:: TOPGEN_1:def 8
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is dense-in-itself
   iff
      [#] b1 is dense-in-itself(b1);

:: TOPGEN_1:th 38
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b1 is being_T1 & b2 is dense-in-itself(b1)
   holds Cl b2 is dense-in-itself(b1);

:: TOPGEN_1:attrnot 4 => TOPGEN_1:attr 4
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is dense-in-itself means
    for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is dense-in-itself(a1);
end;

:: TOPGEN_1:dfs 9
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is dense-in-itself
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is dense-in-itself(a1);

:: TOPGEN_1:def 9
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is dense-in-itself(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 in b2
         holds b3 is dense-in-itself(b1);

:: TOPGEN_1:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is dense-in-itself(b1)
   holds union b2 c= union Der b2;

:: TOPGEN_1:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is dense-in-itself(b1)
   holds union b2 is dense-in-itself(b1);

:: TOPGEN_1:th 41
theorem
for b1 being non empty TopSpace-like TopStruct holds
   Fr {} b1 = {};

:: TOPGEN_1:funcreg 4
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be open closed Element of bool the carrier of a1;
  cluster Fr a2 -> empty;
end;

:: TOPGEN_1:exreg 4
registration
  let a1 be non empty TopSpace-like non discrete TopStruct;
  cluster non open Element of bool the carrier of a1;
end;

:: TOPGEN_1:exreg 5
registration
  let a1 be non empty TopSpace-like non discrete TopStruct;
  cluster non closed Element of bool the carrier of a1;
end;

:: TOPGEN_1:funcreg 5
registration
  let a1 be non empty TopSpace-like non discrete TopStruct;
  let a2 be non open Element of bool the carrier of a1;
  cluster Fr a2 -> non empty;
end;

:: TOPGEN_1:funcreg 6
registration
  let a1 be non empty TopSpace-like non discrete TopStruct;
  let a2 be non closed Element of bool the carrier of a1;
  cluster Fr a2 -> non empty;
end;

:: TOPGEN_1:attrnot 5 => TOPGEN_1:attr 5
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is perfect means
    a2 is closed(a1) & a2 is dense-in-itself(a1);
end;

:: TOPGEN_1:dfs 10
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is perfect
it is sufficient to prove
  thus a2 is closed(a1) & a2 is dense-in-itself(a1);

:: TOPGEN_1:def 10
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is perfect(b1)
   iff
      b2 is closed(b1) & b2 is dense-in-itself(b1);

:: TOPGEN_1:condreg 2
registration
  let a1 be TopSpace-like TopStruct;
  cluster perfect -> closed dense-in-itself (Element of bool the carrier of a1);
end;

:: TOPGEN_1:condreg 3
registration
  let a1 be TopSpace-like TopStruct;
  cluster closed dense-in-itself -> perfect (Element of bool the carrier of a1);
end;

:: TOPGEN_1:th 42
theorem
for b1 being TopSpace-like TopStruct holds
   Der {} b1 = {} b1;

:: TOPGEN_1:th 43
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is perfect(b1)
   iff
      Der b2 = b2;

:: TOPGEN_1:th 44
theorem
for b1 being TopSpace-like TopStruct holds
   {} b1 is perfect(b1);

:: TOPGEN_1:condreg 4
registration
  let a1 be TopSpace-like TopStruct;
  cluster empty -> perfect (Element of bool the carrier of a1);
end;

:: TOPGEN_1:exreg 6
registration
  let a1 be TopSpace-like TopStruct;
  cluster perfect Element of bool the carrier of a1;
end;

:: TOPGEN_1:attrnot 6 => TOPGEN_1:attr 6
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is scattered means
    for b1 being Element of bool the carrier of a1
          st b1 is not empty & b1 c= a2
       holds b1 is not dense-in-itself(a1);
end;

:: TOPGEN_1:dfs 11
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is scattered
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 is not empty & b1 c= a2
       holds b1 is not dense-in-itself(a1);

:: TOPGEN_1:def 11
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is scattered(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 is not empty & b3 c= b2
         holds b3 is not dense-in-itself(b1);

:: TOPGEN_1:condreg 5
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster non empty scattered -> non dense-in-itself (Element of bool the carrier of a1);
end;

:: TOPGEN_1:condreg 6
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster non empty dense-in-itself -> non scattered (Element of bool the carrier of a1);
end;

:: TOPGEN_1:th 45
theorem
for b1 being TopSpace-like TopStruct holds
   {} b1 is scattered(b1);

:: TOPGEN_1:condreg 7
registration
  let a1 be TopSpace-like TopStruct;
  cluster empty -> scattered (Element of bool the carrier of a1);
end;

:: TOPGEN_1:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
      st b1 is being_T1
   holds ex b2, b3 being Element of bool the carrier of b1 st
      b2 \/ b3 = [#] b1 & b2 misses b3 & b2 is perfect(b1) & b3 is scattered(b1);

:: TOPGEN_1:funcreg 7
registration
  let a1 be TopSpace-like discrete TopStruct;
  let a2 be Element of bool the carrier of a1;
  cluster Fr a2 -> empty;
end;

:: TOPGEN_1:condreg 8
registration
  let a1 be TopSpace-like discrete TopStruct;
  cluster -> open closed (Element of bool the carrier of a1);
end;

:: TOPGEN_1:th 47
theorem
for b1 being TopSpace-like discrete TopStruct
for b2 being Element of bool the carrier of b1 holds
   Der b2 = {};

:: TOPGEN_1:funcreg 8
registration
  let a1 be non empty TopSpace-like discrete TopStruct;
  let a2 be Element of bool the carrier of a1;
  cluster Der a2 -> empty;
end;

:: TOPGEN_1:funcnot 4 => TOPGEN_1:func 4
definition
  let a1 be TopSpace-like TopStruct;
  func density A1 -> cardinal set means
    (ex b1 being Element of bool the carrier of a1 st
        b1 is dense(a1) & it = Card b1) &
     (for b1 being Element of bool the carrier of a1
           st b1 is dense(a1)
        holds it c= Card b1);
end;

:: TOPGEN_1:def 12
theorem
for b1 being TopSpace-like TopStruct
for b2 being cardinal set holds
      b2 = density b1
   iff
      (ex b3 being Element of bool the carrier of b1 st
          b3 is dense(b1) & b2 = Card b3) &
       (for b3 being Element of bool the carrier of b1
             st b3 is dense(b1)
          holds b2 c= Card b3);

:: TOPGEN_1:attrnot 7 => TOPGEN_1:attr 7
definition
  let a1 be TopSpace-like TopStruct;
  attr a1 is separable means
    density a1 c= alef 0;
end;

:: TOPGEN_1:dfs 13
definiens
  let a1 be TopSpace-like TopStruct;
To prove
     a1 is separable
it is sufficient to prove
  thus density a1 c= alef 0;

:: TOPGEN_1:def 13
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is separable
   iff
      density b1 c= alef 0;

:: TOPGEN_1:th 49
theorem
for b1 being TopSpace-like countable TopStruct holds
   b1 is separable;

:: TOPGEN_1:condreg 9
registration
  cluster TopSpace-like countable -> separable (TopStruct);
end;

:: TOPGEN_1:th 50
theorem
for b1 being Element of bool the carrier of R^1
      st b1 = RAT
   holds b1 ` = IRRAT;

:: TOPGEN_1:th 51
theorem
for b1 being Element of bool the carrier of R^1
      st b1 = IRRAT
   holds b1 ` = RAT;

:: TOPGEN_1:th 52
theorem
for b1 being Element of bool the carrier of R^1
      st b1 = RAT
   holds Int b1 = {};

:: TOPGEN_1:th 53
theorem
for b1 being Element of bool the carrier of R^1
      st b1 = IRRAT
   holds Int b1 = {};

:: TOPGEN_1:th 54
theorem
for b1 being Element of bool the carrier of R^1
      st b1 = RAT
   holds b1 is dense(R^1);

:: TOPGEN_1:th 55
theorem
for b1 being Element of bool the carrier of R^1
      st b1 = IRRAT
   holds b1 is dense(R^1);

:: TOPGEN_1:th 56
theorem
for b1 being Element of bool the carrier of R^1
      st b1 = RAT
   holds b1 is boundary(R^1);

:: TOPGEN_1:th 57
theorem
for b1 being Element of bool the carrier of R^1
      st b1 = IRRAT
   holds b1 is boundary(R^1);

:: TOPGEN_1:th 58
theorem
for b1 being Element of bool the carrier of R^1
      st b1 = REAL
   holds b1 is boundary(not R^1);

:: TOPGEN_1:th 59
theorem
ex b1, b2 being Element of bool the carrier of R^1 st
   b1 is boundary(R^1) & b2 is boundary(R^1) & b1 \/ b2 is boundary(not R^1);