reserve Y for TopStruct;
reserve X for non empty TopSpace;

theorem Th31:
  for A being Subset of X holds (for x being Point of X st x in A
  ex G being Subset of X st G is open & A /\ G = {x}) implies A is discrete
proof
  let A be Subset of X;
  assume
A1: for x being Point of X st x in A ex G being Subset of X st G is open
  & A /\ G = {x};
  hereby
    per cases;
    suppose
      A is empty;
      hence thesis by Th29;
    end;
    suppose
      A is non empty;
      then consider X0 being strict non empty SubSpace of X such that
A2:   A = the carrier of X0 by TSEP_1:10;
A3:   [#]X = the carrier of X;
      [#]X0 = the carrier of X0;
      then
A4:   the carrier of X0 c= the carrier of X by A3,PRE_TOPC:def 4;
      now
        let C be Subset of X0;
        let y be Point of X0;
        reconsider x = y as Point of X by A4,TARSKI:def 3;
        consider G being Subset of X such that
A5:     G is open and
A6:     A /\ G = {x} by A1,A2;
        assume
A7:     C = {y};
         G in the topology of X & G /\ [#]X0 = C by A2,A7,A6,A5;
        then C in the topology of X0 by PRE_TOPC:def 4;
        hence C is open;
      end;
      then X0 is discrete by TDLAT_3:17;
      hence thesis by A2,Th20;
    end;
  end;
end;
