In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function ${\displaystyle f:X\to Y}$ is open if for any open set ${\displaystyle U}$ in ${\displaystyle X,}$ the image ${\displaystyle f(U)}$ is open in ${\displaystyle Y.}$ Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.

Open and closed maps are not necessarily continuous. Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property; this fact remains true even if one restricts oneself to metric spaces. Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that, by definition, a function ${\displaystyle f:X\to Y}$ is continuous if and only if the preimage of every open set of ${\displaystyle Y}$ is open in ${\displaystyle X.}$ (Equivalently, if and only if the preimage of every closed set of ${\displaystyle Y}$ is closed in ${\displaystyle X}$).

Early study of open maps was pioneered by Simion Stoilow and Gordon Thomas Whyburn.

If ${\displaystyle S}$ is a subset of a topological space then let ${\displaystyle {\overline {S}}}$ and ${\displaystyle \operatorname {Cl} S}$ (resp. ${\displaystyle \operatorname {Int} S}$) denote the closure (resp. interior) of ${\displaystyle S}$ in that space. Let ${\displaystyle f:X\to Y}$ be a function between topological spaces. If ${\displaystyle S}$ is any set then ${\displaystyle f(S):=\left\{f(s)~:~s\in S\cap \operatorname {domain} f\right\}}$ is called the image of ${\displaystyle S}$ under ${\displaystyle f.}$

There are two different competing, but closely related, definitions of "open map" that are widely used, where both of these definitions can be summarized as: "it is a map that sends open sets to open sets." The following terminology is sometimes used to distinguish between the two definitions.

A map ${\displaystyle f:X\to Y}$ is called a

Every strongly open map is a relatively open map. However, these definitions are not equivalent in general.

A surjective map is relatively open if and only if it is strongly open; so for this important special case the definitions are equivalent. More generally, a map ${\displaystyle f:X\to Y}$ is relatively open if and only if the surjection ${\displaystyle f:X\to f(X)}$ is a strongly open map.