reserve a,b,c for set;
reserve r for Real,
  X for set,
  n for Element of NAT;

theorem Th34:
  for X being set, A being Subset of ClFinTop(X) holds A is open
  iff A = {} or A` is finite
proof
  let X be set, A be Subset of ClFinTop(X);
A1: the carrier of ClFinTop X = X by Def6;
  hereby
    assume that
A2: A is open and
A3: A <> {};
    A`` = A;
    then A` <> [#]the carrier of ClFinTop(X) by A3,XBOOLE_1:37;
    hence A` is finite by A2,A1,Def6;
  end;
  assume A = {} or A` is finite;
  then A` is closed by Def6;
  hence thesis by TOPS_1:4;
end;
