for p being Point of T

for V being Subset of Y st (Prj2 (s,H)) . p in V & V is open holds

ex W being Subset of T st

( p in W & W is open & (Prj2 (s,H)) .: W c= V )

for V being Subset of Y st (Prj2 (s,H)) . p in V & V is open holds

ex W being Subset of T st

( p in W & W is open & (Prj2 (s,H)) .: W c= V )

proof

hence
Prj2 (s,H) is continuous
by JGRAPH_2:10; :: thesis: verum
let p be Point of T; :: thesis: for V being Subset of Y st (Prj2 (s,H)) . p in V & V is open holds

ex W being Subset of T st

( p in W & W is open & (Prj2 (s,H)) .: W c= V )

let V be Subset of Y; :: thesis: ( (Prj2 (s,H)) . p in V & V is open implies ex W being Subset of T st

( p in W & W is open & (Prj2 (s,H)) .: W c= V ) )

assume A1: ( (Prj2 (s,H)) . p in V & V is open ) ; :: thesis: ex W being Subset of T st

( p in W & W is open & (Prj2 (s,H)) .: W c= V )

(Prj2 (s,H)) . p = H . (s,p) by Def3;

then consider W being Subset of [:S,T:] such that

A2: [s,p] in W and

A3: W is open and

A4: H .: W c= V by A1, JGRAPH_2:10;

consider A being Subset-Family of [:S,T:] such that

A5: W = union A and

A6: for e being set st e in A holds

ex X1 being Subset of S ex Y1 being Subset of T st

( e = [:X1,Y1:] & X1 is open & Y1 is open ) by A3, BORSUK_1:5;

consider e being set such that

A7: [s,p] in e and

A8: e in A by A2, A5, TARSKI:def 4;

consider X1 being Subset of S, Y1 being Subset of T such that

A9: e = [:X1,Y1:] and

X1 is open and

A10: Y1 is open by A6, A8;

take Y1 ; :: thesis: ( p in Y1 & Y1 is open & (Prj2 (s,H)) .: Y1 c= V )

thus p in Y1 by A7, A9, ZFMISC_1:87; :: thesis: ( Y1 is open & (Prj2 (s,H)) .: Y1 c= V )

thus Y1 is open by A10; :: thesis: (Prj2 (s,H)) .: Y1 c= V

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (Prj2 (s,H)) .: Y1 or x in V )

assume x in (Prj2 (s,H)) .: Y1 ; :: thesis: x in V

then consider c being Point of T such that

A11: c in Y1 and

A12: x = (Prj2 (s,H)) . c by FUNCT_2:65;

s in X1 by A7, A9, ZFMISC_1:87;

then [s,c] in [:X1,Y1:] by A11, ZFMISC_1:87;

then [s,c] in W by A5, A8, A9, TARSKI:def 4;

then A13: H . [s,c] in H .: W by FUNCT_2:35;

(Prj2 (s,H)) . c = H . (s,c) by Def3

.= H . [s,c] ;

hence x in V by A4, A12, A13; :: thesis: verum

end;ex W being Subset of T st

( p in W & W is open & (Prj2 (s,H)) .: W c= V )

let V be Subset of Y; :: thesis: ( (Prj2 (s,H)) . p in V & V is open implies ex W being Subset of T st

( p in W & W is open & (Prj2 (s,H)) .: W c= V ) )

assume A1: ( (Prj2 (s,H)) . p in V & V is open ) ; :: thesis: ex W being Subset of T st

( p in W & W is open & (Prj2 (s,H)) .: W c= V )

(Prj2 (s,H)) . p = H . (s,p) by Def3;

then consider W being Subset of [:S,T:] such that

A2: [s,p] in W and

A3: W is open and

A4: H .: W c= V by A1, JGRAPH_2:10;

consider A being Subset-Family of [:S,T:] such that

A5: W = union A and

A6: for e being set st e in A holds

ex X1 being Subset of S ex Y1 being Subset of T st

( e = [:X1,Y1:] & X1 is open & Y1 is open ) by A3, BORSUK_1:5;

consider e being set such that

A7: [s,p] in e and

A8: e in A by A2, A5, TARSKI:def 4;

consider X1 being Subset of S, Y1 being Subset of T such that

A9: e = [:X1,Y1:] and

X1 is open and

A10: Y1 is open by A6, A8;

take Y1 ; :: thesis: ( p in Y1 & Y1 is open & (Prj2 (s,H)) .: Y1 c= V )

thus p in Y1 by A7, A9, ZFMISC_1:87; :: thesis: ( Y1 is open & (Prj2 (s,H)) .: Y1 c= V )

thus Y1 is open by A10; :: thesis: (Prj2 (s,H)) .: Y1 c= V

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (Prj2 (s,H)) .: Y1 or x in V )

assume x in (Prj2 (s,H)) .: Y1 ; :: thesis: x in V

then consider c being Point of T such that

A11: c in Y1 and

A12: x = (Prj2 (s,H)) . c by FUNCT_2:65;

s in X1 by A7, A9, ZFMISC_1:87;

then [s,c] in [:X1,Y1:] by A11, ZFMISC_1:87;

then [s,c] in W by A5, A8, A9, TARSKI:def 4;

then A13: H . [s,c] in H .: W by FUNCT_2:35;

(Prj2 (s,H)) . c = H . (s,c) by Def3

.= H . [s,c] ;

hence x in V by A4, A12, A13; :: thesis: verum