reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;

theorem Th19:
  W is open implies union W is open
proof
  assume
A1: W is open;
  W c= the topology of GX
  proof
    let x be object;
    assume
A2: x in W;
    then reconsider X=x as Subset of GX;
    X is open by A1,A2;
    hence thesis;
  end;
  then union W in the topology of GX by PRE_TOPC:def 1;
  hence thesis;
end;
