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

theorem Th12:
  for X being non empty TopSpace holds (for A being Subset of X
  holds (A <> the carrier of X implies Int A = {})) implies X is anti-discrete
proof
  let X be non empty TopSpace;
  assume
A1: for A being Subset of X holds (A <> the carrier of X implies Int A = {});
  now
    let A be Subset of X;
    assume A is open;
    then A = Int A by TOPS_1:23;
    hence A = {} or A = the carrier of X by A1;
  end;
  hence thesis by TDLAT_3:18;
end;
