reserve X for TopSpace;

theorem Th16:
  for X0 being SubSpace of X, A being Subset of X st A = the
  carrier of X0 holds X0 is open SubSpace of X iff A is open
proof
  let X0 be SubSpace of X, A be Subset of X;
  assume
A1: A = the carrier of X0;
  hence X0 is open SubSpace of X implies A is open by Def1;
  thus A is open implies X0 is open SubSpace of X
  proof
    assume A is open;
    then
    for B be Subset of X holds B = the carrier of X0 implies B is open by A1;
    hence thesis by Def1;
  end;
end;
