reserve X for TopSpace;

theorem
  for X1 being open SubSpace of X, X2 being open SubSpace of X1 holds X2
  is open SubSpace of X
proof
  let X1 be open SubSpace of X, X2 be open SubSpace of X1;
  now
    reconsider C = [#]X1 as Subset of X by BORSUK_1:1;
    let B be Subset of X;
    assume
A1: B = the carrier of X2;
    then reconsider BB = B as Subset of X1 by BORSUK_1:1;
    BB is open by A1,Def1;
    then
A2: ex A being Subset of X st A is open & A /\ [#]X1 = BB by TOPS_2:24;
    C is open by Def1;
    hence B is open by A2;
  end;
  hence thesis by Def1,Th7;
end;
