reserve X for TopSpace;

theorem
  for X0 being closed SubSpace of X, A being Subset of X, B being Subset
  of X0 st A = B holds B is closed iff A is closed
proof
  let X0 be closed SubSpace of X, A be Subset of X, B be Subset of X0 such
  that
A1: A = B;
  reconsider C = the carrier of X0 as Subset of X by Th1;
  C is closed by Th11;
  hence thesis by A1,Th8;
end;
