reserve X for TopSpace;

theorem Th11:
  for X0 being SubSpace of X, A being Subset of X st A = the
  carrier of X0 holds X0 is closed SubSpace of X iff A is closed
proof
  let X0 be SubSpace of X, A be Subset of X;
  assume
A1: A = the carrier of X0;
  hence X0 is closed SubSpace of X implies A is closed by BORSUK_1:def 11;
  thus A is closed implies X0 is closed SubSpace of X
  proof
    assume A is closed;
    then for B be Subset of X holds B = the carrier of X0 implies B is closed
    by A1;
    hence thesis by BORSUK_1:def 11;
  end;
end;
