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

theorem
  (for A being Subset of X, x being Point of X st A = {x} holds Cl A =
  the carrier of X) implies X is anti-discrete
proof
  assume
A1: for A being Subset of X, x being Point of X st A = {x} holds Cl A =
  the carrier of X;
  for B being Subset of X st B is closed holds B = {} or B = the carrier of X
  proof
    let B be Subset of X;
    assume
A2: B is closed;
    set z = the Element of B;
    assume
A3: B <> {};
    then reconsider x = z as Point of X by TARSKI:def 3;
A4: {x} c= B by A3,ZFMISC_1:31;
    then reconsider A = {x} as Subset of X by XBOOLE_1:1;
    Cl A = the carrier of X by A1;
    then the carrier of X c= B by A2,A4,TOPS_1:5;
    hence thesis;
  end;
  hence thesis by Th19;
end;
