
theorem
  for p being Point of Sierpinski_Space st p = 0 holds {p} is closed
proof
  set S = Sierpinski_Space;
  let p be Point of S;
A1: the carrier of S = {0,1} by WAYBEL18:def 9;
  assume
A2: p = 0;
A3: [#]S \ {p} = {1}
  proof
    thus [#]S \ {p} c= {1}
    proof
      let a be object;
      assume
A4:   a in [#]S \ {p};
      then not a in {p} by XBOOLE_0:def 5;
      then a <> 0 by A2,TARSKI:def 1;
      then a = 1 by A1,A4,TARSKI:def 2;
      hence thesis by TARSKI:def 1;
    end;
    let a be object;
    assume a in {1};
    then
A5: a = 1 by TARSKI:def 1;
    then
A6: not a in {p} by A2,TARSKI:def 1;
    a in [#]S by A1,A5,TARSKI:def 2;
    hence thesis by A6,XBOOLE_0:def 5;
  end;
  the topology of S = {{}, {1}, {0,1}} by WAYBEL18:def 9;
  hence [#]S \ {p} in the topology of S by A3,ENUMSET1:def 1;
end;
