reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;
reserve X for non empty TopSpace;

theorem Th32:
  X is extremally_disconnected iff for A being Subset of X st A is
  condensed holds A is closed & A is open
proof
  thus X is extremally_disconnected implies for A being Subset of X st A is
  condensed holds A is closed & A is open
  proof
    assume
A1: X is extremally_disconnected;
    let A be Subset of X;
A2: Cl Int A is open by A1;
    assume
A3: A is condensed;
    then Cl A = Cl Int A by Th9;
    then Int Cl A = Cl Int A by A2,TOPS_1:23;
    hence thesis by A3,Th8;
  end;
  assume
A4: for A being Subset of X st A is condensed holds A is closed & A is open;
  now
    let A be Subset of X;
    assume A is open;
    then
A5: Int A = A by TOPS_1:23;
    Cl Int A is closed_condensed by TDLAT_1:22;
    then Cl A is condensed by A5,TOPS_1:66;
    hence Cl A is open by A4;
  end;
  hence thesis;
end;
