
theorem
  for T being non empty TopSpace, x,y being Element of T st for W being
  open Subset of T st y in W holds x in W holds (0,1) --> (y,x) is continuous
  Function of Sierpinski_Space, T
proof
  let T be non empty TopSpace;
  let x,y be Element of T such that
A1: for W being open Subset of T st y in W holds x in W;
A2: the carrier of Sierpinski_Space = {0,{0}} by WAYBEL18:def 9, CARD_1:49;
  then reconsider i = (0,{0}) --> (y,x) as Function of Sierpinski_Space, T;
A3: i.1 = x by FUNCT_4:63, CARD_1:49;
A4: not 1 in {0} by TARSKI:def 1;
A5: 0 in {0} by TARSKI:def 1;
A6: 1 in {0,1} by TARSKI:def 2;
A7: i.0 = y by FUNCT_4:63;
A8: now
    let W be Subset of T;
    assume W is open;
    then
A9: y in W & x in W or not y in W & x in W or not y in W & not x in W by A1;
A10:  i"W = {} or i"W = {0} or i"W = {1} or i"W = {0,1}
          by A2,ZFMISC_1:36, CARD_1:49;
    i"W <> {0} by A7,A3,A6,A5,A4,A9,FUNCT_2:38, CARD_1:49;
    then i"W in {{}, {1}, {0,1}}
     by ENUMSET1:def 1,A10;
    then i"W in the topology of Sierpinski_Space by WAYBEL18:def 9;
    hence i"W is open by PRE_TOPC:def 2;
  end;
  [#]T <> {};
  hence thesis by A8,TOPS_2:43, CARD_1:49;
end;
