let T be non empty TopSpace; :: thesis: for M being non empty MetrSpace
for f being Function of (TopSpaceMetr M),T holds
( f is open iff for p being Point of M
for r being positive Real ex W being open Subset of T st
( f . p in W & W c= f .: (Ball (p,r)) ) )
let M be non empty MetrSpace; :: thesis: for f being Function of (TopSpaceMetr M),T holds
( f is open iff for p being Point of M
for r being positive Real ex W being open Subset of T st
( f . p in W & W c= f .: (Ball (p,r)) ) )
let f be Function of (TopSpaceMetr M),T; :: thesis: ( f is open iff for p being Point of M
for r being positive Real ex W being open Subset of T st
( f . p in W & W c= f .: (Ball (p,r)) ) )

hereby :: thesis: ( ( for p being Point of M
for r being positive Real ex W being open Subset of T st
( f . p in W & W c= f .: (Ball (p,r)) ) ) implies f is open )

assume A1: f is open ; :: thesis: for p being Point of M
for r being positive Real ex W being open Subset of T st
( f . p in W & W c= f .: (Ball (p,r)) )
let p be Point of M; :: thesis: for r being positive Real ex W being open Subset of T st
( f . p in W & W c= f .: (Ball (p,r)) )
let r be positive Real; :: thesis: ex W being open Subset of T st
( f . p in W & W c= f .: (Ball (p,r)) )
A2: p in Ball (p,r) by GOBOARD6:1;
reconsider V = Ball (p,r) as Subset of (TopSpaceMetr M) ;
V is open by TOPMETR:14;
then consider W being open Subset of T such that
A3: ( f . p in W & W c= f .: V ) by A1, A2, Th3;
take W = W; :: thesis: ( f . p in W & W c= f .: (Ball (p,r)) )
thus ( f . p in W & W c= f .: (Ball (p,r)) ) by A3; :: thesis: verum

end;

assume A4: for p being Point of M
for r being positive Real ex W being open Subset of T st
( f . p in W & W c= f .: (Ball (p,r)) ) ; :: thesis: f is open
for p being Point of (TopSpaceMetr M)
for V being open Subset of (TopSpaceMetr M) st p in V holds
ex W being open Subset of T st
( f . p in W & W c= f .: V )

proof

let p be Point of (TopSpaceMetr M); :: thesis: for V being open Subset of (TopSpaceMetr M) st p in V holds
ex W being open Subset of T st
( f . p in W & W c= f .: V )
let V be open Subset of (TopSpaceMetr M); :: thesis: ( p in V implies ex W being open Subset of T st
( f . p in W & W c= f .: V ) )
reconsider q = p as Point of M ;
assume p in V ; :: thesis: ex W being open Subset of T st
( f . p in W & W c= f .: V )
then consider r being Real such that
A5: r > 0 and
A6: Ball (q,r) c= V by TOPMETR:15;
consider W being open Subset of T such that
A7: f . p in W and
A8: W c= f .: (Ball (q,r)) by A4, A5;
take W ; :: thesis: ( f . p in W & W c= f .: V )
thus f . p in W by A7; :: thesis: W c= f .: V
f .: (Ball (q,r)) c= f .: V by A6, RELAT_1:123;
hence W c= f .: V by A8; :: thesis: verum

end;

hence f is open by Th3; :: thesis: verum