reserve X for non empty TopSpace,
  D for Subset of X;

theorem Th11:
  for X being non empty TopSpace holds (for A being Subset of X
  holds (A is non empty implies Cl A = the carrier of X)) implies X is
  anti-discrete
proof
  let X be non empty TopSpace;
  assume
A1: for A being Subset of X holds (A is non empty implies Cl A = the
  carrier of X);
  now
    let A be Subset of X;
    assume A is closed;
    then A = Cl A by PRE_TOPC:22;
    hence A = {} or A = the carrier of X by A1;
  end;
  hence thesis by TDLAT_3:19;
end;
