let T be non empty TopSpace; :: thesis: for f being Function of T,R^1 holds
( f is open iff for p being Point of T
for V being open Subset of T st p in V holds
ex r being positive Real st ].((f . p) - r),((f . p) + r).[ c= f .: V )

let f be Function of T,R^1; :: thesis: ( f is open iff for p being Point of T
for V being open Subset of T st p in V holds
ex r being positive Real st ].((f . p) - r),((f . p) + r).[ c= f .: V )

thus ( f is open implies for p being Point of T
for V being open Subset of T st p in V holds
ex r being positive Real st ].((f . p) - r),((f . p) + r).[ c= f .: V ) :: thesis: ( ( for p being Point of T
for V being open Subset of T st p in V holds
ex r being positive Real st ].((f . p) - r),((f . p) + r).[ c= f .: V ) implies f is open )
proof
assume A1: f is open ; :: thesis: for p being Point of T
for V being open Subset of T st p in V holds
ex r being positive Real st ].((f . p) - r),((f . p) + r).[ c= f .: V

let p be Point of T; :: thesis: for V being open Subset of T st p in V holds
ex r being positive Real st ].((f . p) - r),((f . p) + r).[ c= f .: V

let V be open Subset of T; :: thesis: ( p in V implies ex r being positive Real st ].((f . p) - r),((f . p) + r).[ c= f .: V )
assume A2: p in V ; :: thesis: ex r being positive Real st ].((f . p) - r),((f . p) + r).[ c= f .: V
reconsider fp = f . p as Point of RealSpace ;
consider r being positive Real such that
A3: Ball (fp,r) c= f .: V by A1, A2, Th4;
].(fp - r),(fp + r).[ = Ball (fp,r) by FRECHET:7;
hence ex r being positive Real st ].((f . p) - r),((f . p) + r).[ c= f .: V by A3; :: thesis: verum
end;
assume A4: for p being Point of T
for V being open Subset of T st p in V holds
ex r being positive Real st ].((f . p) - r),((f . p) + r).[ c= f .: V ; :: thesis: f is open
for p being Point of T
for V being open Subset of T
for q being Point of RealSpace 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 T; :: thesis: for V being open Subset of T
for q being Point of RealSpace 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 RealSpace 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 RealSpace; :: 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 )
assume p in V ; :: thesis: ex r being positive Real st Ball (q,r) c= f .: V
then consider r being positive Real such that
A6: ].((f . p) - r),((f . p) + r).[ c= f .: V by A4;
].(q - r),(q + r).[ = Ball (q,r) by FRECHET:7;
hence ex r being positive Real st Ball (q,r) c= f .: V by A5, A6; :: thesis: verum
end;
hence f is open by Th4; :: thesis: verum