let m be Nat; :: thesis: for T being non empty TopSpace

for f being Function of T,(TOP-REAL m) holds

( f is continuous iff 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) ) )

let T be non empty TopSpace; :: thesis: for f being Function of T,(TOP-REAL m) holds

( f is continuous iff 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) ) )

let f be Function of T,(TOP-REAL m); :: thesis: ( f is continuous iff 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) ) )

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: ( ( 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) ) ) implies f is continuous )

for r being positive Real ex W being open Subset of T st

( p in W & f .: W c= Ball ((f . p),r) ) ; :: thesis: f is continuous

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 )

for f being Function of T,(TOP-REAL m) holds

( f is continuous iff 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) ) )

let T be non empty TopSpace; :: thesis: for f being Function of T,(TOP-REAL m) holds

( f is continuous iff 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) ) )

let f be Function of T,(TOP-REAL m); :: thesis: ( f is continuous iff 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) ) )

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: ( ( 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) ) ) implies f is continuous )

proof

assume A3:
for p being Point of T
assume A2:
f is continuous
; :: thesis: 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) )

let p be Point of T; :: thesis: for r being positive Real ex W being open Subset of T st

( p in W & f .: W c= Ball ((f . p),r) )

let r be positive Real; :: thesis: ex W being open Subset of T st

( p in W & f .: W c= Ball ((f . p),r) )

f . p in Ball ((f . p),r) by A1, TOPGEN_5:13;

then ex W being Subset of T st

( p in W & W is open & f .: W c= Ball ((f . p),r) ) by A2, JGRAPH_2:10;

hence ex W being open Subset of T st

( p in W & f .: W c= Ball ((f . p),r) ) ; :: thesis: verum

end;for r being positive Real ex W being open Subset of T st

( p in W & f .: W c= Ball ((f . p),r) )

let p be Point of T; :: thesis: for r being positive Real ex W being open Subset of T st

( p in W & f .: W c= Ball ((f . p),r) )

let r be positive Real; :: thesis: ex W being open Subset of T st

( p in W & f .: W c= Ball ((f . p),r) )

f . p in Ball ((f . p),r) by A1, TOPGEN_5:13;

then ex W being Subset of T st

( p in W & W is open & f .: W c= Ball ((f . p),r) ) by A2, JGRAPH_2:10;

hence ex W being open Subset of T st

( p in W & f .: W c= Ball ((f . p),r) ) ; :: thesis: verum

for r being positive Real ex W being open Subset of T st

( p in W & f .: W c= Ball ((f . p),r) ) ; :: thesis: f is continuous

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 )

proof

hence
f is continuous
by JGRAPH_2:10; :: thesis: verum
let p be Point of T; :: thesis: 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 V be Subset of (TOP-REAL m); :: thesis: ( f . p in V & V is open implies ex W being Subset of T st

( p in W & W is open & f .: W c= V ) )

assume A4: f . p in V ; :: thesis: ( not V is open or ex W being Subset of T st

( p in W & W is open & f .: W c= V ) )

reconsider u = f . p as Point of (Euclid m) by EUCLID:67;

assume V is open ; :: thesis: ex W being Subset of T st

( p in W & W is open & f .: W c= V )

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: verum

end;ex W being Subset of T st

( p in W & W is open & f .: W c= V )

let V be Subset of (TOP-REAL m); :: thesis: ( f . p in V & V is open implies ex W being Subset of T st

( p in W & W is open & f .: W c= V ) )

assume A4: f . p in V ; :: thesis: ( not V is open or ex W being Subset of T st

( p in W & W is open & f .: W c= V ) )

reconsider u = f . p as Point of (Euclid m) by EUCLID:67;

assume V is open ; :: thesis: ex W being Subset of T st

( p in W & W is open & f .: W c= V )

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: verum