reserve X for non empty TopSpace;
reserve X for non empty TopSpace;
reserve X for non empty TopSpace,
  X0 for non empty maximal_Kolmogorov_subspace of X;

theorem
  for A being Subset of X holds A is closed implies (Stone-retraction(X,
  X0)).:(A) is closed
proof
  let A be Subset of X;
  reconsider M = the carrier of X0 as Subset of X by TSEP_1:1;
  set B = (Stone-retraction(X,X0)).:(A);
  assume
A1: A is closed;
  then A = MaxADSet(A) by TEX_4:60;
  then A /\ M = B by Def12;
  hence thesis by A1,TSP_1:def 2;
end;
