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

theorem Th4:
  for C being Subset of X modified_with_respect_to D st C = D holds
  D is dense implies C is dense & C is open
proof
  let C be Subset of X modified_with_respect_to D;
  assume
A1: C = D;
  set A = (Cl C)`;
  A in the topology of (X modified_with_respect_to D) by PRE_TOPC:def 2;
  then A in D-extension_of_the_topology_of X by TMAP_1:93;
  then
A2: A in {H \/ (G /\ D) where H is Subset of X, G is Subset of X : H in the
  topology of X & G in the topology of X} by TMAP_1:def 7;
  reconsider B = A as Subset of X by TMAP_1:93;
  consider H, G being Subset of X such that
A3: A = H \/ (G /\ D) and
A4: H in the topology of X and
  G in the topology of X by A2;
A5: H c= A by A3,XBOOLE_1:7;
A6: C c= Cl C by PRE_TOPC:18;
  then D misses A by A1,SUBSET_1:24;
  then (G /\ D) misses A by XBOOLE_1:17,63;
  then
A7: (G /\ D) /\ A = {} by XBOOLE_0:def 7;
  A = (H \/ (G /\ D)) /\ A by A3
    .= (H /\ A) \/ {} by A7,XBOOLE_1:23
    .= H /\ A;
  then A c= H by XBOOLE_1:17;
  then B = H by A5,XBOOLE_0:def 10;
  then
A8: B is open by A4;
  D misses B by A1,A6,SUBSET_1:24;
  then (Cl D) misses B by A8,TSEP_1:36;
  then
A9: (Cl D) /\ B = {} by XBOOLE_0:def 7;
  assume D is dense;
  then
A10: Cl D = [#]X by TOPS_1:def 3;
  thus C is dense
  proof
    assume C is not dense;
    then Cl C <> [#](X modified_with_respect_to D) by TOPS_1:def 3;
    then B <> {}(X modified_with_respect_to D) by TOPS_3:2;
    hence contradiction by A9,A10,XBOOLE_1:28;
  end;
  thus thesis by A1,TMAP_1:94;
end;
