
theorem Th24:
  for Y being non trivial T_0-TopSpace st not Y is T_1 holds
  Sierpinski_Space is_Retract_of Y
proof
  let Y be non trivial T_0-TopSpace;
  given p,q being Point of Y such that
A1: p <> q and
A2: for W,V being Subset of Y st W is open & V is open & p in W & not q
  in W & q in V holds p in V;
  (ex V being Subset of Y st V is open & p in V & not q in V) or ex W
  being Subset of Y st W is open & not p in W & q in W by A1,TSP_1:def 3;
  then consider V being Subset of Y such that
A3: V is open and
A4: p in V & not q in V or not p in V & q in V;
A5: the TopStruct of Omega Y = the TopStruct of Y by WAYBEL25:def 2;
  then consider x,y being Element of Omega Y such that
A6: p in V & not q in V & x = q & y = p or not p in V & q in V & x = p &
  y = q by A4;
  now
    let W be open Subset of Omega Y;
    W is open Subset of Y by A5,TOPS_3:76;
    hence x in W implies y in W by A2,A3,A6;
  end;
  then (0,1) --> (x,y) is continuous Function of Sierpinski_Space, Omega Y by
YELLOW16:47;
  then reconsider
  i = (0,1) --> (x,y) as continuous Function of Sierpinski_Space, Y
  by A5,YELLOW12:36;
  reconsider V as open Subset of Omega Y by A3,A5,TOPS_3:76;
  reconsider c = chi(V, the carrier of Y) as continuous Function of Y,
  Sierpinski_Space by A3,YELLOW16:46;
  c*i = id Sierpinski_Space by A5,A6,YELLOW16:48;
  hence thesis by WAYBEL25:def 1;
end;
