reserve T for TopSpace;

theorem Th9:
  for F being Subset-Family of T holds F = {} iff Cl F = {}
proof
  let F be Subset-Family of T;
  thus F = {} implies Cl F = {} by PCOMPS_1:12;
  assume
A1: Cl F = {};
  set B = the Element of F;
  assume
A2: F <> {};
  then reconsider B as Subset of T by TARSKI:def 3;
  Cl B in Cl F by A2,PCOMPS_1:def 2;
  hence contradiction by A1;
end;
