reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;

theorem Th17:
  (for A being Subset of X, x being Point of X st A = {x} holds A
  is open) implies X is discrete
proof
  assume
A1: for A being Subset of X, x being Point of X st A = {x} holds A is open;
  now
    let A be Subset of X;
    set F = {B where B is Subset of X : ex a being Point of X st a in A & B =
    {a}};
A2: F c= bool A
    proof
      let x be object;
      assume x in F;
      then consider C being Subset of X such that
A3:   x = C and
A4:   ex a being Point of X st a in A & C = {a};
      C c= A by A4,ZFMISC_1:31;
      hence thesis by A3;
    end;
    bool A c= bool the carrier of X by ZFMISC_1:67;
    then reconsider F as Subset-Family of X by A2,XBOOLE_1:1;
A5: union bool A = A by ZFMISC_1:81;
    now
      let x be set;
      assume
A6:   x in bool A;
      then reconsider P = x as Subset of X by XBOOLE_1:1;
      now
        let y be object;
        assume
A7:     y in P;
        then reconsider a = y as Point of X;
        now
          take B0 = {a};
          B0 is Subset of X by A7,ZFMISC_1:31;
          hence y in B0 & B0 in F by A6,A7,TARSKI:def 1;
        end;
        hence y in union F by TARSKI:def 4;
      end;
      hence x c= union F;
    end;
    then
A8: union bool A c= union F;
    now
      let B be Subset of X;
      assume B in F;
      then
      ex C being Subset of X st C = B & ex a being Point of X st a in A &
      C = {a};
      hence B is open by A1;
    end;
    then
A9: F is open by TOPS_2:def 1;
    union F c= union bool A by A2,ZFMISC_1:77;
    then union F = A by A8,A5;
    hence A is open by A9,TOPS_2:19;
  end;
  hence thesis by Th15;
end;
