let m be Nat; :: thesis: for T being non empty TopSpace
for f being Function of (),T holds
( f is continuous iff for p being Point of ()
for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: (Ball (p,s)) c= V )

let T be non empty TopSpace; :: thesis: for f being Function of (),T holds
( f is continuous iff for p being Point of ()
for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: (Ball (p,s)) c= V )

let f be Function of (),T; :: thesis: ( f is continuous iff for p being Point of ()
for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: (Ball (p,s)) c= V )

A1: m in NAT by ORDINAL1:def 12;
hereby :: thesis: ( ( for p being Point of ()
for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: (Ball (p,s)) c= V ) implies f is continuous )
assume A2: f is continuous ; :: thesis: for p being Point of ()
for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: (Ball (p,s)) c= V

let p be Point of (); :: thesis: for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: (Ball (p,s)) c= V

let V be open Subset of T; :: thesis: ( f . p in V implies ex s being positive Real st f .: (Ball (p,s)) c= V )
assume f . p in V ; :: thesis: ex s being positive Real st f .: (Ball (p,s)) c= V
then consider W being Subset of () such that
A3: p in W and
A4: W is open and
A5: f .: W c= V by ;
reconsider u = p as Point of () by EUCLID:67;
Int W = W by ;
then consider s being Real such that
A6: s > 0 and
A7: Ball (u,s) c= W by ;
reconsider s = s as positive Real by A6;
take s = s; :: thesis: f .: (Ball (p,s)) c= V
Ball (u,s) = Ball (p,s) by TOPREAL9:13;
then f .: (Ball (p,s)) c= f .: W by ;
hence f .: (Ball (p,s)) c= V by A5; :: thesis: verum
end;
assume A8: for p being Point of ()
for V being open Subset of T st f . p in V holds
ex s being positive Real st f .: (Ball (p,s)) c= V ; :: thesis: f is continuous
for p being Point of ()
for V being Subset of T st f . p in V & V is open holds
ex W being Subset of () st
( p in W & W is open & f .: W c= V )
proof
let p be Point of (); :: thesis: for V being Subset of T st f . p in V & V is open holds
ex W being Subset of () st
( p in W & W is open & f .: W c= V )

let V be Subset of T; :: thesis: ( f . p in V & V is open implies ex W being Subset of () st
( p in W & W is open & f .: W c= V ) )

assume that
A9: f . p in V and
A10: V is open ; :: thesis: ex W being Subset of () st
( p in W & W is open & f .: W c= V )

consider s being positive Real such that
A11: f .: (Ball (p,s)) c= V by A8, A9, A10;
take W = Ball (p,s); :: thesis: ( p in W & W is open & f .: W c= V )
thus p in W by ; :: thesis: ( W is open & f .: W c= V )
thus W is open ; :: thesis: f .: W c= V
thus f .: W c= V by A11; :: thesis: verum
end;
hence f is continuous by JGRAPH_2:10; :: thesis: verum