
theorem Th19:
  for T being non empty TopSpace holds T is T_1 iff for p being
  Point of T holds {p} is closed
proof
  let T be non empty TopSpace;
  thus T is T_1 implies for p being Point of T holds {p} is closed
  proof
    assume
A1: T is T_1;
    for p being Point of T holds {p} is closed
    proof
      let p be Point of T;
      consider B being Subset of T such that
A2:   B = {p}`;
      defpred Q[Subset of T] means ex q being Point of T st (q in B & for V
      being Subset of T st $1 = V holds (V is open & q in V & not p in V ));
      consider F being Subset-Family of T such that
A3:   for C being Subset of T holds C in F iff Q[C] from SUBSET_1:sch
      3;
A4:   for C being Subset of T holds (C in F iff ex q being Point of T st
      q in B & C is open & q in C & not p in C )
      proof
        let C be Subset of T;
A5:     (ex q being Point of T st (q in B & C is open & q in C & not p in
        C)) implies C in F
        proof
          assume
A6:       ex q being Point of T st q in B & C is open & q in C & not p in C;
          ex q being Point of T st (q in B & for V being Subset of T st C
          = V holds V is open & q in V & not p in V )
          proof
            consider q being Point of T such that
A7:         q in B & C is open & q in C & not p in C by A6;
            take q;
            thus thesis by A7;
          end;
          hence thesis by A3;
        end;
        C in F implies ex q being Point of T st q in B & C is open & q in
        C & not p in C
        proof
          assume C in F;
          then consider q being Point of T such that
A8:       q in B & for V being Subset of T st C = V holds V is open &
          q in V & not p in V by A3;
          take q;
          thus thesis by A8;
        end;
        hence thesis by A5;
      end;
      for x being object holds x in F implies x in the topology of T
      proof
        let x be object;
        assume
A9:     x in F;
        then reconsider x as Subset of T;
        ex q being Point of T st q in B & x is open & q in x & not p in x
        by A4,A9;
        hence thesis;
      end;
      then
A10:  F c= the topology of T;
A11:  for q being Point of T st q in B holds ex V being Subset of T st V
      is open & q in V & not p in V
      proof
        let q be Point of T;
        assume q in B;
        then not q in {p} by A2,XBOOLE_0:def 5;
        then not q = p by TARSKI:def 1;
        then
        ex V,W being Subset of T st V is open & W is open & q in V & not p
        in V & p in W & not q in W by A1;
        then consider V being Subset of T such that
A12:    V is open & q in V & not p in V;
        take V;
        thus thesis by A12;
      end;
      for x being object holds x in B implies x in union F
      proof
        let x be object;
        assume
A13:    x in B;
        then reconsider x as Point of T;
        consider C being Subset of T such that
A14:    C is open & x in C & not p in C by A11,A13;
        ex C being set st x in C & C in F
        by A4,A13,A14;
        hence thesis by TARSKI:def 4;
      end;
      then
A15:  B c= union F;
      for x being object holds x in union F implies x in B
      proof
        let x be object;
        assume x in union F;
        then consider C being set such that
A16:    x in C and
A17:    C in F by TARSKI:def 4;
        reconsider C as Subset of T by A17;
        ex q being Point of T st q in B & C is open & q in C & not p in C
        by A4,A17;
        then C c= [#](T) \ {p} by ZFMISC_1:34;
        hence thesis by A2,A16;
      end;
      then union F c= B;
      then B = union F by A15;
      then B in the topology of T by A10,PRE_TOPC:def 1;
      then {p}` = [#](T) \ {p} & B is open;
      hence thesis by A2;
    end;
    hence thesis;
  end;
  assume
A18: for p being Point of T holds {p} is closed;
  for p,q being Point of T st not p = q ex W,V being Subset of T st W is
  open & V is open & p in W & not q in W & q in V & not p in V
  proof
    let p,q be Point of T;
    consider V,W being Subset of T such that
A19: V = {p}` and
A20: W = {q}`;
    p in {p} by TARSKI:def 1;
    then
A21: not p in V by A19,XBOOLE_0:def 5;
    assume
A22: not p = q;
    then not p in {q} by TARSKI:def 1;
    then
A23: p in W by A20,XBOOLE_0:def 5;
    q in {q} by TARSKI:def 1;
    then
A24: not q in W by A20,XBOOLE_0:def 5;
    not q in {p} by A22,TARSKI:def 1;
    then
A25: q in V by A19,XBOOLE_0:def 5;
    {p} is closed & {q} is closed by A18;
    hence thesis by A19,A20,A23,A24,A25,A21;
  end;
  hence thesis;
end;
