let T be non empty TopSpace; :: thesis: for M being non empty MetrSpace

for f being Function of T,(TopSpaceMetr M) holds

( f is open iff for p being Point of T

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V )

let M be non empty MetrSpace; :: thesis: for f being Function of T,(TopSpaceMetr M) holds

( f is open iff for p being Point of T

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V )

let f be Function of T,(TopSpaceMetr M); :: thesis: ( f is open iff for p being Point of T

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V )

thus ( f is open implies for p being Point of T

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V ) :: thesis: ( ( for p being Point of T

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V ) implies f is open )

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V ; :: thesis: f is open

for p being Point of T

for V being open Subset of T st p in V holds

ex W being open Subset of (TopSpaceMetr M) st

( f . p in W & W c= f .: V )

for f being Function of T,(TopSpaceMetr M) holds

( f is open iff for p being Point of T

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V )

let M be non empty MetrSpace; :: thesis: for f being Function of T,(TopSpaceMetr M) holds

( f is open iff for p being Point of T

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V )

let f be Function of T,(TopSpaceMetr M); :: thesis: ( f is open iff for p being Point of T

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V )

thus ( f is open implies for p being Point of T

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V ) :: thesis: ( ( for p being Point of T

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V ) implies f is open )

proof

assume A5:
for p being Point of T
assume A1:
f is open
; :: thesis: for p being Point of T

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V

let p be Point of T; :: thesis: for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V

let V be open Subset of T; :: thesis: for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V

let q be Point of M; :: thesis: ( q = f . p & p in V implies ex r being positive Real st Ball (q,r) c= f .: V )

assume A2: q = f . p ; :: thesis: ( not p in V or ex r being positive Real st Ball (q,r) c= f .: V )

assume p in V ; :: thesis: ex r being positive Real st Ball (q,r) c= f .: V

then consider W being open Subset of (TopSpaceMetr M) such that

A3: f . p in W and

A4: W c= f .: V by A1, Th3;

ex r being Real st

( r > 0 & Ball (q,r) c= W ) by A2, A3, TOPMETR:15;

hence ex r being positive Real st Ball (q,r) c= f .: V by A4, XBOOLE_1:1; :: thesis: verum

end;for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V

let p be Point of T; :: thesis: for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V

let V be open Subset of T; :: thesis: for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V

let q be Point of M; :: thesis: ( q = f . p & p in V implies ex r being positive Real st Ball (q,r) c= f .: V )

assume A2: q = f . p ; :: thesis: ( not p in V or ex r being positive Real st Ball (q,r) c= f .: V )

assume p in V ; :: thesis: ex r being positive Real st Ball (q,r) c= f .: V

then consider W being open Subset of (TopSpaceMetr M) such that

A3: f . p in W and

A4: W c= f .: V by A1, Th3;

ex r being Real st

( r > 0 & Ball (q,r) c= W ) by A2, A3, TOPMETR:15;

hence ex r being positive Real st Ball (q,r) c= f .: V by A4, XBOOLE_1:1; :: thesis: verum

for V being open Subset of T

for q being Point of M st q = f . p & p in V holds

ex r being positive Real st Ball (q,r) c= f .: V ; :: thesis: f is open

for p being Point of T

for V being open Subset of T st p in V holds

ex W being open Subset of (TopSpaceMetr M) st

( f . p in W & W c= f .: V )

proof

hence
f is open
by Th3; :: thesis: verum
let p be Point of T; :: thesis: for V being open Subset of T st p in V holds

ex W being open Subset of (TopSpaceMetr M) st

( f . p in W & W c= f .: V )

let V be open Subset of T; :: thesis: ( p in V implies ex W being open Subset of (TopSpaceMetr M) st

( f . p in W & W c= f .: V ) )

reconsider q = f . p as Point of M ;

assume p in V ; :: thesis: ex W being open Subset of (TopSpaceMetr M) st

( f . p in W & W c= f .: V )

then consider r being positive Real such that

A6: Ball (q,r) c= f .: V by A5;

reconsider W = Ball (q,r) as open Subset of (TopSpaceMetr M) by TOPMETR:14;

take W ; :: thesis: ( f . p in W & W c= f .: V )

thus f . p in W by GOBOARD6:1; :: thesis: W c= f .: V

thus W c= f .: V by A6; :: thesis: verum

end;ex W being open Subset of (TopSpaceMetr M) st

( f . p in W & W c= f .: V )

let V be open Subset of T; :: thesis: ( p in V implies ex W being open Subset of (TopSpaceMetr M) st

( f . p in W & W c= f .: V ) )

reconsider q = f . p as Point of M ;

assume p in V ; :: thesis: ex W being open Subset of (TopSpaceMetr M) st

( f . p in W & W c= f .: V )

then consider r being positive Real such that

A6: Ball (q,r) c= f .: V by A5;

reconsider W = Ball (q,r) as open Subset of (TopSpaceMetr M) by TOPMETR:14;

take W ; :: thesis: ( f . p in W & W c= f .: V )

thus f . p in W by GOBOARD6:1; :: thesis: W c= f .: V

thus W c= f .: V by A6; :: thesis: verum