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 Th20:
  for r being continuous Function of X,X0 for M being Subset of X
st M = the carrier of X0 holds (for a being Point of X holds M /\ MaxADSet(a) =
  {r.a}) implies r is being_a_retraction
proof
  let r be continuous Function of X,X0;
  let M be Subset of X;
  reconsider N = M as Subset of X;
  assume
A1: M = the carrier of X0;
  then N is maximal_T_0 by Th11;
  then
A2: N is T_0;
  assume
A3: for a being Point of X holds M /\ MaxADSet(a) = {r.a};
  for x being Point of X st x in the carrier of X0 holds r.x = x
  proof
    let x be Point of X;
    assume x in the carrier of X0;
    then
A4: M /\ MaxADSet(x) = {x} by A1,A2;
    M /\ MaxADSet(x) = {r.x} by A3;
    hence thesis by A4,ZFMISC_1:3;
  end;
  hence thesis by BORSUK_1:def 16;
end;
