let X, Y be non empty TopSpace; :: thesis: for f being Function of X,Y holds
( f is open iff for p being Point of X
for V being open Subset of X st p in V holds
ex W being open Subset of Y st
( f . p in W & W c= f .: V ) )

let f be Function of X,Y; :: thesis: ( f is open iff for p being Point of X
for V being open Subset of X st p in V holds
ex W being open Subset of Y st
( f . p in W & W c= f .: V ) )

thus ( f is open implies for p being Point of X
for V being open Subset of X st p in V holds
ex W being open Subset of Y st
( f . p in W & W c= f .: V ) ) :: thesis: ( ( for p being Point of X
for V being open Subset of X st p in V holds
ex W being open Subset of Y st
( f . p in W & W c= f .: V ) ) implies f is open )
proof
assume A1: f is open ; :: thesis: for p being Point of X
for V being open Subset of X st p in V holds
ex W being open Subset of Y st
( f . p in W & W c= f .: V )

let p be Point of X; :: thesis: for V being open Subset of X st p in V holds
ex W being open Subset of Y st
( f . p in W & W c= f .: V )

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

assume A2: p in V ; :: thesis: ex W being open Subset of Y st
( f . p in W & W c= f .: V )

V is a_neighborhood of p by ;
then consider W being open a_neighborhood of f . p such that
A3: W c= f .: V by ;
take W ; :: thesis: ( f . p in W & W c= f .: V )
thus f . p in W by CONNSP_2:4; :: thesis: W c= f .: V
thus W c= f .: V by A3; :: thesis: verum
end;
assume A4: for p being Point of X
for V being open Subset of X st p in V holds
ex W being open Subset of Y st
( f . p in W & W c= f .: V ) ; :: thesis: f is open
for p being Point of X
for P being open a_neighborhood of p ex R being a_neighborhood of f . p st R c= f .: P
proof
let p be Point of X; :: thesis: for P being open a_neighborhood of p ex R being a_neighborhood of f . p st R c= f .: P
let P be open a_neighborhood of p; :: thesis: ex R being a_neighborhood of f . p st R c= f .: P
consider W being open Subset of Y such that
A5: f . p in W and
A6: W c= f .: P by ;
W is a_neighborhood of f . p by ;
hence ex R being a_neighborhood of f . p st R c= f .: P by A6; :: thesis: verum
end;
hence f is open by TOPGRP_1:23; :: thesis: verum