let M1, M2 be non empty MetrSpace; :: thesis: for f being Function of (TopSpaceMetr M1),(TopSpaceMetr M2) holds
( f is open iff for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) )
let f be Function of (TopSpaceMetr M1),(TopSpaceMetr M2); :: thesis: ( f is open iff for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) )
thus ( f is open implies for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) ) :: thesis: ( ( for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) ) implies f is open )

proof

assume A1: f is open ; :: thesis: for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r))
let p be Point of M1; :: thesis: for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r))
let q be Point of M2; :: thesis: for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r))
let r be positive Real; :: thesis: ( q = f . p implies ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) )
assume A2: q = f . p ; :: thesis: ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r))
reconsider V = Ball (p,r) as Subset of (TopSpaceMetr M1) ;
A3: p in V by GOBOARD6:1;
V is open by TOPMETR:14;
hence ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) by A1, A2, A3, Th4; :: thesis: verum

end;

assume A4: for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) ; :: thesis: f is open
for p being Point of (TopSpaceMetr M1)
for V being open Subset of (TopSpaceMetr M1)
for q being Point of M2 st q = f . p & p in V holds
ex r being positive Real st Ball (q,r) c= f .: V

proof

let p be Point of (TopSpaceMetr M1); :: thesis: for V being open Subset of (TopSpaceMetr M1)
for q being Point of M2 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 (TopSpaceMetr M1); :: thesis: for q being Point of M2 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 M2; :: thesis: ( q = f . p & p in V implies ex r being positive Real st Ball (q,r) c= f .: V )
assume A5: q = f . p ; :: thesis: ( not p in V or ex r being positive Real st Ball (q,r) c= f .: V )
reconsider p1 = p as Point of M1 ;
assume p in V ; :: thesis: ex r being positive Real st Ball (q,r) c= f .: V
then consider r being Real such that
A6: r > 0 and
A7: Ball (p1,r) c= V by TOPMETR:15;
A8: f .: (Ball (p1,r)) c= f .: V by A7, RELAT_1:123;
ex s being positive Real st Ball (q,s) c= f .: (Ball (p1,r)) by A4, A5, A6;
hence ex r being positive Real st Ball (q,r) c= f .: V by A8, XBOOLE_1:1; :: thesis: verum

end;

hence f is open by Th4; :: thesis: verum