reserve X for non empty TopSpace;
reserve X for non empty TopSpace;

theorem Th14:
  for X0 being maximal_Kolmogorov_subspace of X for G being Subset
of X, G0 being Subset of X0 st G0 = G holds G0 is open iff MaxADSet(G) is open
  & G0 = MaxADSet(G) /\ the carrier of X0
proof
  let X0 be maximal_Kolmogorov_subspace of X;
  let G be Subset of X, G0 be Subset of X0;
  reconsider M = the carrier of X0 as Subset of X by Lm1;
  assume
A1: G0 = G;
A2: M is maximal_T_0 by Th11;
  thus G0 is open implies MaxADSet(G) is open & G0 = MaxADSet(G) /\ the
  carrier of X0
  proof
    assume G0 is open;
    then
A3: ex H being Subset of X st H is open & G0 = H /\ M by TSP_1:def 1;
    hence MaxADSet(G) is open by A2,A1,Th6;
    thus thesis by A2,A1,A3,Th6;
  end;
  assume
A4: MaxADSet(G) is open;
  assume G0 = MaxADSet(G) /\ the carrier of X0;
  hence thesis by A4,TSP_1:def 1;
end;
