nLab closed map

This page is about the concept in topology. For the more general concept see at closed morphism.

Definition

Definition

(open maps and closed maps)

A continuous function $f \colon (X,\tau_X) \to (Y, \tau_Y)$ is called

• an open map if the image under $f$ of an open subset of $X$ is an open subset of $Y$;

• a closed map if the image under $f$ of a closed subset of $X$ (def. ) is a closed subset of $Y$.

Examples

Example

(image projections of open/closed maps are themselves open/closed)

If a continuous function $f \colon (X,\tau_X) \to (Y,\tau_Y)$ is an open map or closed map (def. ) then so its its image projection $X \to f(X) \subset Y$, respectively, for $f(X) \subset Y$ regarded with its subspace topology.

Proof

If $f$ is an open map, and $O \subset X$ is an open subset, so that $f(O) \subset Y$ is also open in $Y$, then, since $f(O) = f(O) \cap f(X)$, it is also still open in the subspace topology, hence $X \to f(X)$ is an open map.

If $f$ is a closed map, and $C \subset X$ is a closed subset so that also $f(C) \subset Y$ is a closed subset, then the complement $Y \backslash f(C)$ is open in $Y$ and hence $(Y \backslash f(C)) \cap f(X) = f(X) \backslash f(C)$ is open in the subspace topology, which means that $f(C)$ is closed in the subspace topology.

Proposition

(maps from compact spaces to Hausdorff spaces are closed and proper)

Let $f \colon (X, \tau_X) \longrightarrow (Y, \tau_Y)$ be a continuous function between topological spaces such that

1. $(X,\tau_X)$ is a compact topological space;

2. $(Y,\tau_Y)$ is a Hausdorff topological space.

Then $f$ is

1. a closed map (def. );

2. a proper map.

Proposition

(proper maps to locally compact spaces are closed)

Let

1. $(X,\tau_X)$ be a topological space,

2. $(Y,\tau_Y)$ a locally compact topological space according to def. ,

3. $f \colon X \to Y$ a continuous function.

Then:

If $f$ is a proper map, then it is a closed map.

Related concepts

• closed morphism

• embedding of topological spaces

• open map, proper map