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

theorem Th1:
  for B being Subset of X, C being Subset of X
  modified_with_respect_to D st B = C holds B is open implies C is open
proof
  let B be Subset of X, C be Subset of X modified_with_respect_to D;
  assume
A1: B = C;
A2: the topology of X c= D-extension_of_the_topology_of X by TMAP_1:88;
A3: the topology of X modified_with_respect_to D = D
  -extension_of_the_topology_of X by TMAP_1:93;
  assume B in the topology of X;
  hence thesis by A1,A2,A3;
end;
