reserve X for set;

theorem
  for T being TopSpace, F being Subset-Family of T holds F is open iff F
  is Subset of InclPoset the topology of T
proof
  let T be TopSpace;
  let F be Subset-Family of T;
  hereby
    assume
A1: F is open;
    F c= the topology of T
    by A1,PRE_TOPC:def 2;
    hence F is Subset of InclPoset the topology of T;
  end;
  assume
A2: F is Subset of InclPoset the topology of T;
  let P be Subset of T;
  assume P in F;
  hence thesis by A2,PRE_TOPC:def 2;
end;
