Open and Closed Mappings

We are almost ready to prove the open mapping theorems. We first need to define what it means for a map to be open and for a map to be closed.

Definition: Let $X$ and $Y$ be topological spaces. A function $f : X \to Y$ is said to be Open or an Open Mapping if for every open set $U$ in $X$ we have that the image, $f(U)$ is an open set in $Y$. A function $f : X \to Y$ is said to be Closed or a Closed Mappng if for every closed set $C$ in $X$ we have that the image, $f(C)$ is a closed set in $Y$.

There is an important subtlety to make. Recall that a function $f : X \to Y$ is said to be continuous on $X$ if for every open set $V$ in $Y$ we have that the inverse image $f^{-1}(V)$ is an open set in $X$ (or equivalently, for every closed set $D$ in $Y$ we have that the inverse image $f^{-1}(D)$ is a closed set in $X$). As you can see, there appears to be a connection between open/closed mappings and continuous functions - and there is, but only when $f$ is bijective. We state this result below.

Theorem 1: Let $X$ and $Y$ be topological spaces and let $f : X \to Y$ be bijective. Then: a) $f$ is an open mapping if and only if $f^{-1}$ is continuous. b) $f$ is a closed mapping if and only if $f^{-1}$ is continuous.

Now recall that $f : X \to Y$ is said to be a homeomorphism between $X$ and $Y$ if $f$ is bijective, $f$ is continuous, and $f^{-1}$ is continuous. Thus:

Theorem 2: Let $X$ and $Y$ be topological and let $f : X \to Y$. Then $f$ is a homeomorphism from $X$ to $Y$ if and only if $f$ is a bijective, continuous, and open mapping.

Of course, often times we will NOT be looking at bijective maps. The result above is nevertheless important to remember in the setting when $f$ is bijective though.