Article COMPTS_1, MML version 4.99.1005

:: COMPTS_1:prednot 1 => COMPTS_1:pred 1
definition
  let a1 be 1-sorted;
  let a2 be Element of bool bool the carrier of a1;
  let a3 be Element of bool the carrier of a1;
  pred A2 is_a_cover_of A3 means
    a3 c= union a2;
end;

:: COMPTS_1:dfs 1
definiens
  let a1 be 1-sorted;
  let a2 be Element of bool bool the carrier of a1;
  let a3 be Element of bool the carrier of a1;
To prove
     a2 is_a_cover_of a3
it is sufficient to prove
  thus a3 c= union a2;

:: COMPTS_1:def 1
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b2 is_a_cover_of b3
   iff
      b3 c= union b2;

:: COMPTS_1:attrnot 1 => COMPTS_1:attr 1
definition
  let a1 be set;
  attr a1 is centered means
    a1 <> {} &
     (for b1 being set
           st b1 <> {} & b1 c= a1 & b1 is finite
        holds meet b1 <> {});
end;

:: COMPTS_1:dfs 2
definiens
  let a1 be set;
To prove
     a1 is centered
it is sufficient to prove
  thus a1 <> {} &
     (for b1 being set
           st b1 <> {} & b1 c= a1 & b1 is finite
        holds meet b1 <> {});

:: COMPTS_1:def 2
theorem
for b1 being set holds
      b1 is centered
   iff
      b1 <> {} &
       (for b2 being set
             st b2 <> {} & b2 c= b1 & b2 is finite
          holds meet b2 <> {});

:: COMPTS_1:attrnot 2 => COMPTS_1:attr 2
definition
  let a1 be TopStruct;
  attr a1 is compact means
    for b1 being Element of bool bool the carrier of a1
          st b1 is_a_cover_of a1 & b1 is open(a1)
       holds ex b2 being Element of bool bool the carrier of a1 st
          b2 c= b1 & b2 is_a_cover_of a1 & b2 is finite;
end;

:: COMPTS_1:dfs 3
definiens
  let a1 be TopStruct;
To prove
     a1 is compact
it is sufficient to prove
  thus for b1 being Element of bool bool the carrier of a1
          st b1 is_a_cover_of a1 & b1 is open(a1)
       holds ex b2 being Element of bool bool the carrier of a1 st
          b2 c= b1 & b2 is_a_cover_of a1 & b2 is finite;

:: COMPTS_1:def 3
theorem
for b1 being TopStruct holds
      b1 is compact
   iff
      for b2 being Element of bool bool the carrier of b1
            st b2 is_a_cover_of b1 & b2 is open(b1)
         holds ex b3 being Element of bool bool the carrier of b1 st
            b3 c= b2 & b3 is_a_cover_of b1 & b3 is finite;

:: COMPTS_1:attrnot 3 => COMPTS_1:attr 3
definition
  let a1 be TopStruct;
  attr a1 is being_T2 means
    for b1, b2 being Element of the carrier of a1
          st b1 <> b2
       holds ex b3, b4 being Element of bool the carrier of a1 st
          b3 is open(a1) & b4 is open(a1) & b1 in b3 & b2 in b4 & b3 misses b4;
end;

:: COMPTS_1:dfs 4
definiens
  let a1 be TopStruct;
To prove
     a1 is being_T2
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
          st b1 <> b2
       holds ex b3, b4 being Element of bool the carrier of a1 st
          b3 is open(a1) & b4 is open(a1) & b1 in b3 & b2 in b4 & b3 misses b4;

:: COMPTS_1:def 4
theorem
for b1 being TopStruct holds
      b1 is being_T2
   iff
      for b2, b3 being Element of the carrier of b1
            st b2 <> b3
         holds ex b4, b5 being Element of bool the carrier of b1 st
            b4 is open(b1) & b5 is open(b1) & b2 in b4 & b3 in b5 & b4 misses b5;

:: COMPTS_1:attrnot 4 => COMPTS_1:attr 4
definition
  let a1 be TopStruct;
  attr a1 is being_T3 means
    for b1 being Element of the carrier of a1
    for b2 being Element of bool the carrier of a1
          st b2 <> {} & b2 is closed(a1) & b1 in b2 `
       holds ex b3, b4 being Element of bool the carrier of a1 st
          b3 is open(a1) & b4 is open(a1) & b1 in b3 & b2 c= b4 & b3 misses b4;
end;

:: COMPTS_1:dfs 5
definiens
  let a1 be TopStruct;
To prove
     a1 is being_T3
it is sufficient to prove
  thus for b1 being Element of the carrier of a1
    for b2 being Element of bool the carrier of a1
          st b2 <> {} & b2 is closed(a1) & b1 in b2 `
       holds ex b3, b4 being Element of bool the carrier of a1 st
          b3 is open(a1) & b4 is open(a1) & b1 in b3 & b2 c= b4 & b3 misses b4;

:: COMPTS_1:def 5
theorem
for b1 being TopStruct holds
      b1 is being_T3
   iff
      for b2 being Element of the carrier of b1
      for b3 being Element of bool the carrier of b1
            st b3 <> {} & b3 is closed(b1) & b2 in b3 `
         holds ex b4, b5 being Element of bool the carrier of b1 st
            b4 is open(b1) & b5 is open(b1) & b2 in b4 & b3 c= b5 & b4 misses b5;

:: COMPTS_1:attrnot 5 => COMPTS_1:attr 5
definition
  let a1 be TopStruct;
  attr a1 is being_T4 means
    for b1, b2 being Element of bool the carrier of a1
          st b1 <> {} & b2 <> {} & b1 is closed(a1) & b2 is closed(a1) & b1 misses b2
       holds ex b3, b4 being Element of bool the carrier of a1 st
          b3 is open(a1) & b4 is open(a1) & b1 c= b3 & b2 c= b4 & b3 misses b4;
end;

:: COMPTS_1:dfs 6
definiens
  let a1 be TopStruct;
To prove
     a1 is being_T4
it is sufficient to prove
  thus for b1, b2 being Element of bool the carrier of a1
          st b1 <> {} & b2 <> {} & b1 is closed(a1) & b2 is closed(a1) & b1 misses b2
       holds ex b3, b4 being Element of bool the carrier of a1 st
          b3 is open(a1) & b4 is open(a1) & b1 c= b3 & b2 c= b4 & b3 misses b4;

:: COMPTS_1:def 6
theorem
for b1 being TopStruct holds
      b1 is being_T4
   iff
      for b2, b3 being Element of bool the carrier of b1
            st b2 <> {} & b3 <> {} & b2 is closed(b1) & b3 is closed(b1) & b2 misses b3
         holds ex b4, b5 being Element of bool the carrier of b1 st
            b4 is open(b1) & b5 is open(b1) & b2 c= b4 & b3 c= b5 & b4 misses b5;

:: COMPTS_1:attrnot 6 => COMPTS_1:attr 3
notation
  let a1 be TopStruct;
  synonym Hausdorff for being_T2;
end;

:: COMPTS_1:prednot 2 => COMPTS_1:attr 3
notation
  let a1 be TopStruct;
  synonym a1 is_T2 for being_T2;
end;

:: COMPTS_1:prednot 3 => COMPTS_1:attr 4
notation
  let a1 be TopStruct;
  synonym a1 is_T3 for being_T3;
end;

:: COMPTS_1:prednot 4 => COMPTS_1:attr 5
notation
  let a1 be TopStruct;
  synonym a1 is_T4 for being_T4;
end;

:: COMPTS_1:attrnot 7 => COMPTS_1:attr 6
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is compact means
    for b1 being Element of bool bool the carrier of a1
          st b1 is_a_cover_of a2 & b1 is open(a1)
       holds ex b2 being Element of bool bool the carrier of a1 st
          b2 c= b1 & b2 is_a_cover_of a2 & b2 is finite;
end;

:: COMPTS_1:dfs 7
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is compact
it is sufficient to prove
  thus for b1 being Element of bool bool the carrier of a1
          st b1 is_a_cover_of a2 & b1 is open(a1)
       holds ex b2 being Element of bool bool the carrier of a1 st
          b2 c= b1 & b2 is_a_cover_of a2 & b2 is finite;

:: COMPTS_1:def 7
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is compact(b1)
   iff
      for b3 being Element of bool bool the carrier of b1
            st b3 is_a_cover_of b2 & b3 is open(b1)
         holds ex b4 being Element of bool bool the carrier of b1 st
            b4 c= b3 & b4 is_a_cover_of b2 & b4 is finite;

:: COMPTS_1:th 9
theorem
for b1 being TopStruct holds
   {} b1 is compact(b1);

:: COMPTS_1:th 10
theorem
for b1 being TopStruct holds
      b1 is compact
   iff
      [#] b1 is compact(b1);

:: COMPTS_1:th 11
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
      st b3 c= [#] b2
   holds    b3 is compact(b1)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 = b3
         holds b4 is compact(b2);

:: COMPTS_1:th 12
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   (b2 = {} implies    (b2 is compact(b1)
    iff
       b1 | b2 is compact)) &
    (b1 is TopSpace-like & b2 <> {} implies    (b2 is compact(b1)
    iff
       b1 | b2 is compact));

:: COMPTS_1:th 13
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is compact
   iff
      for b2 being Element of bool bool the carrier of b1
            st b2 is centered & b2 is closed(b1)
         holds meet b2 <> {};

:: COMPTS_1:th 14
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is compact
   iff
      for b2 being Element of bool bool the carrier of b1
            st b2 <> {} & b2 is closed(b1) & meet b2 = {}
         holds ex b3 being Element of bool bool the carrier of b1 st
            b3 <> {} & b3 c= b2 & b3 is finite & meet b3 = {};

:: COMPTS_1:th 15
theorem
for b1 being TopSpace-like TopStruct
   st b1 is being_T2
for b2 being Element of bool the carrier of b1
   st b2 <> {} & b2 is compact(b1)
for b3 being Element of the carrier of b1
      st b3 in b2 `
   holds ex b4, b5 being Element of bool the carrier of b1 st
      b4 is open(b1) & b5 is open(b1) & b3 in b4 & b2 c= b5 & b4 misses b5;

:: COMPTS_1:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b1 is being_T2 & b2 is compact(b1)
   holds b2 is closed(b1);

:: COMPTS_1:th 17
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
      st b1 is compact & b2 is closed(b1)
   holds b2 is compact(b1);

:: COMPTS_1:th 18
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is compact(b1) & b3 c= b2 & b3 is closed(b1)
   holds b3 is compact(b1);

:: COMPTS_1:th 19
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is compact(b1) & b3 is compact(b1)
   holds b2 \/ b3 is compact(b1);

:: COMPTS_1:th 20
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b1 is being_T2 & b2 is compact(b1) & b3 is compact(b1)
   holds b2 /\ b3 is compact(b1);

:: COMPTS_1:th 21
theorem
for b1 being TopSpace-like TopStruct
      st b1 is being_T2 & b1 is compact
   holds b1 is being_T3;

:: COMPTS_1:th 22
theorem
for b1 being TopSpace-like TopStruct
      st b1 is being_T2 & b1 is compact
   holds b1 is being_T4;

:: COMPTS_1:th 23
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 b1 is compact & b3 is continuous(b1, b2) & rng b3 = [#] b2
   holds b2 is compact;

:: COMPTS_1:th 24
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
      st b4 is continuous(b1, b3) & rng b4 = [#] b3 & b2 is compact(b1)
   holds b4 .: b2 is compact(b3);

:: COMPTS_1:th 25
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
   st b1 is compact & b2 is being_T2 & rng b3 = [#] b2 & b3 is continuous(b1, b2)
for b4 being Element of bool the carrier of b1
      st b4 is closed(b1)
   holds b3 .: b4 is closed(b2);

:: COMPTS_1:th 26
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
      st b1 is compact & b2 is being_T2 & dom b3 = [#] b1 & rng b3 = [#] b2 & b3 is one-to-one & b3 is continuous(b1, b2)
   holds b3 is being_homeomorphism(b1, b2);

:: COMPTS_1:funcnot 1 => COMPTS_1:func 1
definition
  let a1 be set;
  func 1TopSp A1 -> TopStruct equals
    TopStruct(#a1,[#] bool a1#);
end;

:: COMPTS_1:def 8
theorem
for b1 being set holds
   1TopSp b1 = TopStruct(#b1,[#] bool b1#);

:: COMPTS_1:funcreg 1
registration
  let a1 be set;
  cluster 1TopSp a1 -> strict TopSpace-like;
end;

:: COMPTS_1:funcreg 2
registration
  let a1 be non empty set;
  cluster 1TopSp a1 -> non empty;
end;

:: COMPTS_1:funcreg 3
registration
  let a1 be set;
  cluster 1TopSp {a1} -> being_T2;
end;

:: COMPTS_1:exreg 1
registration
  cluster non empty TopSpace-like being_T2 TopStruct;
end;

:: COMPTS_1:condreg 1
registration
  let a1 be non empty TopSpace-like being_T2 TopStruct;
  cluster compact -> closed (Element of bool the carrier of a1);
end;

:: COMPTS_1:exreg 2
registration
  let a1 be 1-sorted;
  cluster finite Element of bool the carrier of a1;
end;

:: COMPTS_1:th 27
theorem
for b1 being TopSpace-like TopStruct
      st the carrier of b1 is finite
   holds b1 is compact;

:: COMPTS_1:condreg 2
registration
  let a1 be TopSpace-like TopStruct;
  cluster finite -> compact (Element of bool the carrier of a1);
end;