Article TDLAT_3, MML version 4.99.1005

:: TDLAT_3:th 2
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Cl b2 = (Int (b2 `)) `;

:: TDLAT_3:th 3
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Cl (b2 `) = (Int b2) `;

:: TDLAT_3:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Int (b2 `) = (Cl b2) `;

:: TDLAT_3:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 \/ b3 = the carrier of b1
   holds (b2 is closed(b1) implies b2 \/ Int b3 = the carrier of b1) &
    (b3 is closed(b1) implies (Int b2) \/ b3 = the carrier of b1);

:: TDLAT_3:th 7
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
      Cl b2 = Int b2;

:: TDLAT_3:th 8
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is open(b1) & b2 is closed(b1)
   holds Int Cl b2 = Cl Int b2;

:: TDLAT_3:th 9
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is condensed(b1) & Cl Int b2 c= Int Cl b2
   holds b2 is open_condensed(b1) & b2 is closed_condensed(b1);

:: TDLAT_3:th 10
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is condensed(b1) & Cl Int b2 c= Int Cl b2
   holds b2 is open(b1) & b2 is closed(b1);

:: TDLAT_3:th 11
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is condensed(b1)
   holds Int Cl b2 = Int b2 & Cl b2 = Cl Int b2;

:: TDLAT_3:attrnot 1 => TDLAT_3:attr 1
definition
  let a1 be TopStruct;
  attr a1 is discrete means
    the topology of a1 = bool the carrier of a1;
end;

:: TDLAT_3:dfs 1
definiens
  let a1 be TopStruct;
To prove
     a1 is discrete
it is sufficient to prove
  thus the topology of a1 = bool the carrier of a1;

:: TDLAT_3:def 1
theorem
for b1 being TopStruct holds
      b1 is discrete
   iff
      the topology of b1 = bool the carrier of b1;

:: TDLAT_3:attrnot 2 => TDLAT_3:attr 2
definition
  let a1 be TopStruct;
  attr a1 is anti-discrete means
    the topology of a1 = {{},the carrier of a1};
end;

:: TDLAT_3:dfs 2
definiens
  let a1 be TopStruct;
To prove
     a1 is anti-discrete
it is sufficient to prove
  thus the topology of a1 = {{},the carrier of a1};

:: TDLAT_3:def 2
theorem
for b1 being TopStruct holds
      b1 is anti-discrete
   iff
      the topology of b1 = {{},the carrier of b1};

:: TDLAT_3:th 12
theorem
for b1 being TopStruct
      st b1 is discrete & b1 is anti-discrete
   holds bool the carrier of b1 = {{},the carrier of b1};

:: TDLAT_3:th 13
theorem
for b1 being TopStruct
      st {} in the topology of b1 &
         the carrier of b1 in the topology of b1 &
         bool the carrier of b1 = {{},the carrier of b1}
   holds b1 is discrete & b1 is anti-discrete;

:: TDLAT_3:exreg 1
registration
  cluster non empty strict discrete anti-discrete TopStruct;
end;

:: TDLAT_3:th 14
theorem
for b1 being discrete TopStruct
for b2 being Element of bool the carrier of b1 holds
   (the carrier of b1) \ b2 in the topology of b1;

:: TDLAT_3:th 15
theorem
for b1 being anti-discrete TopStruct
for b2 being Element of bool the carrier of b1
      st b2 in the topology of b1
   holds (the carrier of b1) \ b2 in the topology of b1;

:: TDLAT_3:condreg 1
registration
  cluster discrete -> TopSpace-like (TopStruct);
end;

:: TDLAT_3:condreg 2
registration
  cluster anti-discrete -> TopSpace-like (TopStruct);
end;

:: TDLAT_3:th 16
theorem
for b1 being TopSpace-like TopStruct
      st bool the carrier of b1 = {{},the carrier of b1}
   holds b1 is discrete & b1 is anti-discrete;

:: TDLAT_3:attrnot 3 => TDLAT_3:attr 3
definition
  let a1 be TopStruct;
  attr a1 is almost_discrete means
    for b1 being Element of bool the carrier of a1
          st b1 in the topology of a1
       holds (the carrier of a1) \ b1 in the topology of a1;
end;

:: TDLAT_3:dfs 3
definiens
  let a1 be TopStruct;
To prove
     a1 is almost_discrete
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 in the topology of a1
       holds (the carrier of a1) \ b1 in the topology of a1;

:: TDLAT_3:def 3
theorem
for b1 being TopStruct holds
      b1 is almost_discrete
   iff
      for b2 being Element of bool the carrier of b1
            st b2 in the topology of b1
         holds (the carrier of b1) \ b2 in the topology of b1;

:: TDLAT_3:condreg 3
registration
  cluster discrete -> almost_discrete (TopStruct);
end;

:: TDLAT_3:condreg 4
registration
  cluster anti-discrete -> almost_discrete (TopStruct);
end;

:: TDLAT_3:exreg 2
registration
  cluster strict almost_discrete TopStruct;
end;

:: TDLAT_3:exreg 3
registration
  cluster non empty strict TopSpace-like discrete anti-discrete TopStruct;
end;

:: TDLAT_3:th 17
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is discrete
   iff
      for b2 being Element of bool the carrier of b1 holds
         b2 is open(b1);

:: TDLAT_3:th 18
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is discrete
   iff
      for b2 being Element of bool the carrier of b1 holds
         b2 is closed(b1);

:: TDLAT_3:th 19
theorem
for b1 being TopSpace-like TopStruct
      st for b2 being Element of bool the carrier of b1
        for b3 being Element of the carrier of b1
              st b2 = {b3}
           holds b2 is open(b1)
   holds b1 is discrete;

:: TDLAT_3:condreg 5
registration
  let a1 be non empty TopSpace-like discrete TopStruct;
  cluster -> closed open discrete (SubSpace of a1);
end;

:: TDLAT_3:exreg 4
registration
  let a1 be non empty TopSpace-like discrete TopStruct;
  cluster strict TopSpace-like closed open discrete almost_discrete SubSpace of a1;
end;

:: TDLAT_3:th 20
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is anti-discrete
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is open(b1) & b2 <> {}
         holds b2 = the carrier of b1;

:: TDLAT_3:th 21
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is anti-discrete
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is closed(b1) & b2 <> {}
         holds b2 = the carrier of b1;

:: TDLAT_3:th 22
theorem
for b1 being TopSpace-like TopStruct
      st for b2 being Element of bool the carrier of b1
        for b3 being Element of the carrier of b1
              st b2 = {b3}
           holds Cl b2 = the carrier of b1
   holds b1 is anti-discrete;

:: TDLAT_3:condreg 6
registration
  let a1 be non empty TopSpace-like anti-discrete TopStruct;
  cluster -> anti-discrete (SubSpace of a1);
end;

:: TDLAT_3:exreg 5
registration
  let a1 be non empty TopSpace-like anti-discrete TopStruct;
  cluster TopSpace-like anti-discrete almost_discrete SubSpace of a1;
end;

:: TDLAT_3:th 23
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is almost_discrete
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is open(b1)
         holds b2 is closed(b1);

:: TDLAT_3:th 24
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is almost_discrete
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is closed(b1)
         holds b2 is open(b1);

:: TDLAT_3:th 25
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is almost_discrete
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is open(b1)
         holds Cl b2 = b2;

:: TDLAT_3:th 26
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is almost_discrete
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is closed(b1)
         holds Int b2 = b2;

:: TDLAT_3:exreg 6
registration
  cluster strict TopSpace-like almost_discrete TopStruct;
end;

:: TDLAT_3:th 27
theorem
for b1 being TopSpace-like TopStruct
      st for b2 being Element of bool the carrier of b1
        for b3 being Element of the carrier of b1
              st b2 = {b3}
           holds Cl b2 is open(b1)
   holds b1 is almost_discrete;

:: TDLAT_3:th 28
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is discrete
   iff
      b1 is almost_discrete &
       (for b2 being Element of bool the carrier of b1
       for b3 being Element of the carrier of b1
             st b2 = {b3}
          holds b2 is closed(b1));

:: TDLAT_3:condreg 7
registration
  cluster TopSpace-like discrete -> almost_discrete (TopStruct);
end;

:: TDLAT_3:condreg 8
registration
  cluster TopSpace-like anti-discrete -> almost_discrete (TopStruct);
end;

:: TDLAT_3:condreg 9
registration
  let a1 be non empty TopSpace-like almost_discrete TopStruct;
  cluster non empty -> almost_discrete (SubSpace of a1);
end;

:: TDLAT_3:condreg 10
registration
  let a1 be non empty TopSpace-like almost_discrete TopStruct;
  cluster open -> closed (SubSpace of a1);
end;

:: TDLAT_3:condreg 11
registration
  let a1 be non empty TopSpace-like almost_discrete TopStruct;
  cluster closed -> open (SubSpace of a1);
end;

:: TDLAT_3:exreg 7
registration
  let a1 be non empty TopSpace-like almost_discrete TopStruct;
  cluster non empty strict TopSpace-like almost_discrete SubSpace of a1;
end;

:: TDLAT_3:attrnot 4 => TDLAT_3:attr 4
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is extremally_disconnected means
    for b1 being Element of bool the carrier of a1
          st b1 is open(a1)
       holds Cl b1 is open(a1);
end;

:: TDLAT_3:dfs 4
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is extremally_disconnected
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 is open(a1)
       holds Cl b1 is open(a1);

:: TDLAT_3:def 4
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is open(b1)
         holds Cl b2 is open(b1);

:: TDLAT_3:exreg 8
registration
  cluster non empty strict TopSpace-like extremally_disconnected TopStruct;
end;

:: TDLAT_3:th 29
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is closed(b1)
         holds Int b2 is closed(b1);

:: TDLAT_3:th 30
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      for b2, b3 being Element of bool the carrier of b1
            st b2 is open(b1) & b3 is open(b1) & b2 misses b3
         holds Cl b2 misses Cl b3;

:: TDLAT_3:th 31
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      for b2, b3 being Element of bool the carrier of b1
            st b2 is closed(b1) & b3 is closed(b1) & b2 \/ b3 = the carrier of b1
         holds (Int b2) \/ Int b3 = the carrier of b1;

:: TDLAT_3:th 32
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is open(b1)
         holds Cl b2 = Int Cl b2;

:: TDLAT_3:th 33
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is closed(b1)
         holds Int b2 = Cl Int b2;

:: TDLAT_3:th 34
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is condensed(b1)
         holds b2 is closed(b1) & b2 is open(b1);

:: TDLAT_3:th 35
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is condensed(b1)
         holds b2 is closed_condensed(b1) & b2 is open_condensed(b1);

:: TDLAT_3:th 36
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is condensed(b1)
         holds Int Cl b2 = Cl Int b2;

:: TDLAT_3:th 37
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is condensed(b1)
         holds Int b2 = Cl b2;

:: TDLAT_3:th 38
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      for b2 being Element of bool the carrier of b1 holds
            b2 is open_condensed(b1)
         iff
            b2 is closed_condensed(b1);

:: TDLAT_3:attrnot 5 => TDLAT_3:attr 5
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is hereditarily_extremally_disconnected means
    for b1 being non empty SubSpace of a1 holds
       b1 is extremally_disconnected;
end;

:: TDLAT_3:dfs 5
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is hereditarily_extremally_disconnected
it is sufficient to prove
  thus for b1 being non empty SubSpace of a1 holds
       b1 is extremally_disconnected;

:: TDLAT_3:def 5
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is hereditarily_extremally_disconnected
   iff
      for b2 being non empty SubSpace of b1 holds
         b2 is extremally_disconnected;

:: TDLAT_3:exreg 9
registration
  cluster non empty strict TopSpace-like hereditarily_extremally_disconnected TopStruct;
end;

:: TDLAT_3:condreg 12
registration
  cluster non empty TopSpace-like hereditarily_extremally_disconnected -> extremally_disconnected (TopStruct);
end;

:: TDLAT_3:condreg 13
registration
  cluster non empty TopSpace-like almost_discrete -> hereditarily_extremally_disconnected (TopStruct);
end;

:: TDLAT_3:th 39
theorem
for b1 being non empty TopSpace-like extremally_disconnected TopStruct
for b2 being non empty SubSpace of b1
for b3 being Element of bool the carrier of b1
      st b3 = the carrier of b2 & b3 is dense(b1)
   holds b2 is extremally_disconnected;

:: TDLAT_3:condreg 14
registration
  let a1 be non empty TopSpace-like extremally_disconnected TopStruct;
  cluster non empty open -> extremally_disconnected (SubSpace of a1);
end;

:: TDLAT_3:exreg 10
registration
  let a1 be non empty TopSpace-like extremally_disconnected TopStruct;
  cluster non empty strict TopSpace-like extremally_disconnected SubSpace of a1;
end;

:: TDLAT_3:condreg 15
registration
  let a1 be non empty TopSpace-like hereditarily_extremally_disconnected TopStruct;
  cluster non empty -> hereditarily_extremally_disconnected (SubSpace of a1);
end;

:: TDLAT_3:exreg 11
registration
  let a1 be non empty TopSpace-like hereditarily_extremally_disconnected TopStruct;
  cluster non empty strict TopSpace-like extremally_disconnected hereditarily_extremally_disconnected SubSpace of a1;
end;

:: TDLAT_3:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2 being non empty closed SubSpace of b1 holds
           b2 is extremally_disconnected
   holds b1 is hereditarily_extremally_disconnected;

:: TDLAT_3:th 41
theorem
for b1 being non empty TopSpace-like extremally_disconnected TopStruct holds
   Domains_of b1 = Closed_Domains_of b1;

:: TDLAT_3:th 42
theorem
for b1 being non empty TopSpace-like extremally_disconnected TopStruct holds
   Domains_Union b1 = Closed_Domains_Union b1 & Domains_Meet b1 = Closed_Domains_Meet b1;

:: TDLAT_3:th 43
theorem
for b1 being non empty TopSpace-like extremally_disconnected TopStruct holds
   Domains_Lattice b1 = Closed_Domains_Lattice b1;

:: TDLAT_3:th 44
theorem
for b1 being non empty TopSpace-like extremally_disconnected TopStruct holds
   Domains_of b1 = Open_Domains_of b1;

:: TDLAT_3:th 45
theorem
for b1 being non empty TopSpace-like extremally_disconnected TopStruct holds
   Domains_Union b1 = Open_Domains_Union b1 & Domains_Meet b1 = Open_Domains_Meet b1;

:: TDLAT_3:th 46
theorem
for b1 being non empty TopSpace-like extremally_disconnected TopStruct holds
   Domains_Lattice b1 = Open_Domains_Lattice b1;

:: TDLAT_3:th 47
theorem
for b1 being non empty TopSpace-like extremally_disconnected TopStruct
for b2, b3 being Element of Domains_of b1 holds
(Domains_Union b1) .(b2,b3) = b2 \/ b3 & (Domains_Meet b1) .(b2,b3) = b2 /\ b3;

:: TDLAT_3:th 48
theorem
for b1 being non empty TopSpace-like extremally_disconnected TopStruct
for b2, b3 being Element of the carrier of Domains_Lattice b1
for b4, b5 being Element of Domains_of b1
      st b2 = b4 & b3 = b5
   holds b2 "\/" b3 = b4 \/ b5 & b2 "/\" b3 = b4 /\ b5;

:: TDLAT_3:th 49
theorem
for b1 being non empty TopSpace-like extremally_disconnected TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is domains-family(b1)
for b3 being Element of bool the carrier of Domains_Lattice b1
      st b3 = b2
   holds "\/"(b3,Domains_Lattice b1) = Cl union b2;

:: TDLAT_3:th 50
theorem
for b1 being non empty TopSpace-like extremally_disconnected TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is domains-family(b1)
for b3 being Element of bool the carrier of Domains_Lattice b1
      st b3 = b2
   holds (b3 = {} or "/\"(b3,Domains_Lattice b1) = Int meet b2) &
    (b3 = {} implies "/\"(b3,Domains_Lattice b1) = [#] b1);

:: TDLAT_3:th 51
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      Domains_Lattice b1 is non empty Lattice-like modular LattStr;

:: TDLAT_3:th 52
theorem
for b1 being non empty TopSpace-like TopStruct
      st Domains_Lattice b1 = Closed_Domains_Lattice b1
   holds b1 is extremally_disconnected;

:: TDLAT_3:th 53
theorem
for b1 being non empty TopSpace-like TopStruct
      st Domains_Lattice b1 = Open_Domains_Lattice b1
   holds b1 is extremally_disconnected;

:: TDLAT_3:th 54
theorem
for b1 being non empty TopSpace-like TopStruct
      st Closed_Domains_Lattice b1 = Open_Domains_Lattice b1
   holds b1 is extremally_disconnected;

:: TDLAT_3:th 55
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is extremally_disconnected
   iff
      Domains_Lattice b1 is non empty Lattice-like Boolean LattStr;