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

theorem Th16:
  for X0 being maximal_Kolmogorov_subspace of X for F being Subset
  of X, F0 being Subset of X0 st F0 = F holds F0 is closed iff MaxADSet(F) is
  closed & F0 = MaxADSet(F) /\ the carrier of X0
proof
  let X0 be maximal_Kolmogorov_subspace of X;
  reconsider M = the carrier of X0 as Subset of X by TSEP_1:1;
  let F be Subset of X, F0 be Subset of X0;
  assume
A1: F0 = F;
A2: M is maximal_T_0 by Th11;
  thus F0 is closed implies MaxADSet(F) is closed & F0 = MaxADSet(F) /\ the
  carrier of X0
  proof
    assume F0 is closed;
    then
A3: ex H being Subset of X st H is closed & F0 = H /\ M by TSP_1:def 2;
    hence MaxADSet(F) is closed by A2,A1,Th5;
    thus thesis by A2,A1,A3,Th5;
  end;
  assume
A4: MaxADSet(F) is closed;
  assume F0 = MaxADSet(F) /\ the carrier of X0;
  hence thesis by A4,TSP_1:def 2;
end;
