let S, T be non empty TopSpace; :: thesis: for f being Function of S,T st f is one-to-one & f is onto holds
( f is open iff f " is continuous )
let f be Function of S,T; :: thesis: ( f is one-to-one & f is onto implies ( f is open iff f " is continuous ) )
assume A1: f is one-to-one ; :: thesis: ( not f is onto or ( f is open iff f " is continuous ) )
assume f is onto ; :: thesis: ( f is open iff f " is continuous )
then A2: rng f = [#] T ;
then rng (f ") = [#] S by A1, TOPS_2:49;
then A3: f " is onto ;
( f " is one-to-one & (f ") " = f ) by A1, A2, TOPS_2:50, TOPS_2:51;
hence ( f is open iff f " is continuous ) by A3, Th13; :: thesis: verum