f is an open map iff f is a closed map.

Question

Let X and Y be metric spaces and suppose that f:X --> Y is one one and onto. Show that f is an open map iff it is a closed map. CLOSED MAP-a function which takes closed sets onto closed sets. This problem is from functions of one complex variable by John B Conway.

Answer 1

Let $f$ be given to be an open map .

To show that $f$ is closed.

Let $V$ be closed in $X\implies V^c\text{ is open in X}\implies f(V^c)\text{is open in Y}$

Since $f$ is bijective $f(X\setminus V)=Y\setminus f(V)$

So $f(V^c)\text{is open in Y}\implies Y\setminus f(V)$ is open in $Y\implies f(V)$ is closed in $Y$.

Apply the same logic to show that $f$ is open map given $f$ to be a closed map.