
theorem Th5:
  for X being non empty TopSpace ex f being Function of InclPoset
  the topology of X, oContMaps(X, Sierpinski_Space) st f is isomorphic & for V
  being open Subset of X holds f.V = chi(V, the carrier of X)
proof
  let X be non empty TopSpace;
  deffunc F(set) = chi($1, the carrier of X);
  consider f being Function such that
A1: dom f = the topology of X and
A2: for a being set st a in the topology of X holds f.a = F(a) from
  FUNCT_1:sch 5;
A3: rng f c= the carrier of oContMaps(X, Sierpinski_Space)
  proof
    let x be object;
    assume x in rng f;
    then consider a being object such that
A4: a in dom f and
A5: x = f.a by FUNCT_1:def 3;
    reconsider a as Subset of X by A1,A4;
    a is open by A1,A4,PRE_TOPC:def 2;
    then
    chi(a, the carrier of X) is continuous Function of X, Sierpinski_Space
    by YELLOW16:46;
    then x is continuous Function of X, Sierpinski_Space by A1,A2,A4,A5;
    then x is Element of oContMaps(X, Sierpinski_Space) by Th2;
    hence thesis;
  end;
  set S = InclPoset the topology of X;
A6: the carrier of InclPoset the topology of X = the topology of X by
YELLOW_1:1;
  then reconsider f as Function of InclPoset the topology of X, oContMaps(X,
  Sierpinski_Space) by A1,A3,FUNCT_2:2;
A7: now
    let x,y be Element of S;
    x in the topology of X & y in the topology of X by A6;
    then reconsider V = x, W = y as open Subset of X by PRE_TOPC:def 2;
    set cx = chi(V, the carrier of X), cy = chi(W, the carrier of X);
A8: f.x = cx & f.y = cy by A2,A6;
    cx is continuous Function of X, Sierpinski_Space by YELLOW16:46;
    then
A9: cx is Element of oContMaps(X,Sierpinski_Space) by Th2;
    cy is continuous Function of X, Sierpinski_Space by YELLOW16:46;
    then cy is Element of oContMaps(X,Sierpinski_Space) by Th2;
    then reconsider
    cx, cy as continuous Function of X, Omega Sierpinski_Space by A9,Th1;
    x <= y iff V c= W by YELLOW_1:3;
    then x <= y iff cx <= cy by Th4,YELLOW16:49;
    hence x <= y iff f.x <= f.y by A8,Th3;
  end;
  set T = oContMaps(X, Sierpinski_Space);
A10: rng f = the carrier of T
  proof
    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 V = {1} as open Subset of Sierpinski_Space by
PRE_TOPC:def 2;
    thus rng f c= the carrier of T;
    let t be object;
    assume t in the carrier of T;
    then reconsider g = t as continuous Function of X, Sierpinski_Space by Th2;
    [#]Sierpinski_Space <> {};
    then
A11: g"V is open by TOPS_2:43;
    then reconsider c = chi(g"V, the carrier of X) as Function of X,
    Sierpinski_Space by YELLOW16:46;
    now
      let x be Element of X;
      x in g"V or not x in g"V;
      then
A12:  g.x in V & c.x = 1 or not g.x in V & c.x = 0 by FUNCT_2:38,FUNCT_3:def 3;
      the carrier of Sierpinski_Space = {0,1} by WAYBEL18:def 9;
      then g.x = 0 or g.x = 1 by TARSKI:def 2;
      hence g.x = c.x by A12,TARSKI:def 1;
    end;
    then
A13: g = c by FUNCT_2:63;
A14: g"V in the topology of X by A11,PRE_TOPC:def 2;
    then f.(g"V) = chi(g"V, the carrier of X) by A2;
    hence thesis by A1,A14,A13,FUNCT_1:def 3;
  end;
  take f;
  f is one-to-one
  proof
    let x,y be Element of S;
    x in the topology of X & y in the topology of X by A6;
    then reconsider V = x, W = y as Subset of X;
    f.x = chi(V, the carrier of X) & f.y = chi(W, the carrier of X) by A2,A6;
    hence thesis by FUNCT_3:38;
  end;
  hence f is isomorphic by A10,A7,WAYBEL_0:66;
  let V be open Subset of X;
  V in the topology of X by PRE_TOPC:def 2;
  hence thesis by A2;
end;
