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

theorem Th22:
  for A being Subset of STS(D,d0) holds (A c= D \ {d0} implies A
  is open) & (A <> D & A is open implies A c= D \ {d0})
proof
  let A be Subset of STS(D,d0);
  set Z = A`;
  reconsider P = {d0} as Subset of STS(D,d0);
  thus A c= D \ {d0} implies A is open
  proof
    assume A c= D \ {d0};
    then [#]STS(D,d0) \ (D \ {d0}) c= [#]STS(D,d0) \ A by XBOOLE_1:34;
    then P c= Z by PRE_TOPC:3;
    then Z is closed by Th20;
    hence thesis by TOPS_1:4;
  end;
  thus A <> D & A is open implies A c= D \ {d0}
  proof
    assume A <> D;
    then
A1: Z <> {}STS(D,d0) by TOPS_3:2;
    assume A is open;
    then {d0} c= Z by A1,Th20;
    then Z` c= P` by SUBSET_1:12;
    hence thesis;
  end;
end;
