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

theorem Th19:
  X is anti-discrete iff for A being Subset of X st A is closed
  holds A = {} or A = the carrier of X
proof
  thus X is anti-discrete implies for A being Subset of X st A is closed holds
  A = {} or A = the carrier of X
  proof
    assume
A1: X is anti-discrete;
    let A be Subset of X;
    assume A is closed;
    then A` = {} or A` = the carrier of X by A1,Th18;
    then A`` = [#]X or A`` = {}X by XBOOLE_1:37;
    hence thesis;
  end;
  assume
A2: for A being Subset of X st A is closed holds A = {} or A = the
  carrier of X;
  now
    let B be Subset of X;
    assume B is open;
    then B` = {} or B` = the carrier of X by A2;
    then B`` = [#]X or B`` = {}X by XBOOLE_1:37;
    hence B = {} or B = the carrier of X;
  end;
  hence thesis by Th18;
end;
