
theorem
  for X being non empty TopSpace for V being open Subset of X holds (
  alpha X)".V = chi(V, the carrier of X)
proof
A1: the carrier of Sierpinski_Space = {0, 1} by WAYBEL18:def 9;
  the topology of Sierpinski_Space = {{}, {1}, {0,1}} by WAYBEL18:def 9;
  then {1} in the topology of Sierpinski_Space by ENUMSET1:def 1;
  then reconsider A = {1} as open Subset of Sierpinski_Space by PRE_TOPC:def 2;
  let X be non empty TopSpace;
  consider f be Function of InclPoset the topology of X, oContMaps(X,
  Sierpinski_Space) such that
A2: f is isomorphic and
A3: for V being open Subset of X holds f.V = chi(V, the carrier of X) by
WAYBEL26:5;
A4: the carrier of InclPoset the topology of X = the topology of X by
YELLOW_1:1;
A5: rng f = [#]oContMaps(X, Sierpinski_Space) by A2,WAYBEL_0:66;
A6: f" = f qua Function" by A2,TOPS_2:def 4;
  now
    let x be Element of oContMaps(X, Sierpinski_Space);
    reconsider g = x as continuous Function of X, Sierpinski_Space by
WAYBEL26:2;
    [#]Sierpinski_Space <> {};
    then
A7: g"A is open by TOPS_2:43;
    then
A8: g"A in the topology of X;
A9: f.(g"A) = chi(g"A, the carrier of X) by A3,A7
      .= x by A1,FUNCT_3:40;
    thus (alpha X).x = g"A by Def4
      .= f".x by A2,A6,A4,A8,A9,FUNCT_2:26;
  end;
  then alpha X = f" by FUNCT_2:63;
  then (alpha X)" = f by A2,A5,TOPS_2:51;
  hence thesis by A3;
end;
