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; :: thesis: for V being open Subset of X holds ((alpha X) ") . V = chi (V, the carrier of X)

consider f being 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 " by A2, TOPS_2:def 4;

then (alpha X) " = f by A2, A5, TOPS_2:51;

hence for V being open Subset of X holds ((alpha X) ") . V = chi (V, the carrier of X) by A3; :: thesis: verum

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; :: thesis: for V being open Subset of X holds ((alpha X) ") . V = chi (V, the carrier of X)

consider f being 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 " by A2, TOPS_2:def 4;

now :: thesis: for x being Element of (oContMaps (X,Sierpinski_Space)) holds (alpha X) . x = (f ") . x

then
alpha X = f "
by FUNCT_2:63;let x be Element of (oContMaps (X,Sierpinski_Space)); :: thesis: (alpha X) . x = (f ") . x

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 ; :: thesis: verum

end;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 ; :: thesis: verum

then (alpha X) " = f by A2, A5, TOPS_2:51;

hence for V being open Subset of X holds ((alpha X) ") . V = chi (V, the carrier of X) by A3; :: thesis: verum