reserve X for non empty TopSpace,
  D for Subset of X;

theorem
  for B being Subset of X, C being Subset of X modified_with_respect_to
  D st B = C holds B is closed implies C is closed
proof
  let B be Subset of X, C be Subset of X modified_with_respect_to D;
  assume
A1: B = C;
A2: the carrier of (X modified_with_respect_to D) = the carrier of X by
TMAP_1:93;
  assume B is closed;
  then C` is open by A1,A2,Th1;
  hence thesis;
end;
