reserve X for non empty TopSpace,
  D for Subset of X;
reserve D for non empty set,
  d0 for Element of D;

theorem Th20:
  for A being Subset of STS(D,d0) holds ({d0} c= A implies A is
  closed) & (A is non empty & A is closed implies {d0} c= A)
proof
  let A be Subset of STS(D,d0);
  set Z = A`;
  set G = {P where P is Subset of D : d0 in P & P <> D};
  thus {d0} c= A implies A is closed
  proof
A1: d0 in {d0} by TARSKI:def 1;
    assume {d0} c= A;
    then not ex Q being Subset of D st Q = Z & d0 in Q & Q <> D by A1,
XBOOLE_0:def 5;
    then not Z in G;
    then Z in the topology of STS(D,d0) by XBOOLE_0:def 5;
    then Z is open;
    hence thesis;
  end;
  assume A is non empty;
  then
A2: Z <> D by TOPS_3:1;
  assume A is closed;
  then Z in the topology of STS(D,d0) by PRE_TOPC:def 2;
  then not Z in G by XBOOLE_0:def 5;
  then not d0 in Z by A2;
  then d0 in A by SUBSET_1:29;
  hence thesis by ZFMISC_1:31;
end;
