let m be Nat; :: thesis:
let T be non empty TopSpace; :: thesis:
let f be Function of T,(TOP-REAL m); :: thesis:
A1: m in NAT by ORDINAL1:def 12;
thus ( f is continuous implies for p being Point of T
for r being positive Real ex W being open Subset of T st
( p in W & f .: W c= Ball ((f . p),r) ) ) :: thesis:

end;

assume A3: for p being Point of T
for r being positive Real ex W being open Subset of T st
( p in W & f .: W c= Ball ((f . p),r) ) ; :: thesis:
for p being Point of T
for V being Subset of (TOP-REAL m) st f . p in V & V is open holds
ex W being Subset of T st
( p in W & W is open & f .: W c= V )

let p be Point of T; :: thesis:
let V be Subset of (TOP-REAL m); :: thesis:
assume A4: f . p in V ; :: thesis:
reconsider u = f . p as Point of (Euclid m) by EUCLID:67;
assume V is open ; :: thesis:
then Int V = V by TOPS_1:23;
then consider e being Real such that
A5: e > 0 and
A6: Ball (u,e) c= V by A4, GOBOARD6:5;
A7: Ball (u,e) = Ball ((f . p),e) by TOPREAL9:13;
ex W being open Subset of T st
( p in W & f .: W c= Ball ((f . p),e) ) by A3, A5;
hence ex W being Subset of T st
( p in W & W is open & f .: W c= V ) by A6, A7, XBOOLE_1:1; :: thesis:

end;