# nLab open map

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

# Contents

## Definition

###### Definition

(open maps and closed maps)

A continuous function $f \colon (X,\tau_X) \to (Y, \tau_Y)$ between topological spaces 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 $X$ 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. 1) then so is 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.

###### Example

(projections out of product spaces are open maps)

For $(X_1,\tau_{X_1})$ and $(X_2,\tau_{X_2})$ two topological spaces, then the projection maps

$pr_i \;\colon\; (X_1 \times X_2, \tau_{X_1 \times X_2}) \longrightarrow (X_i, \tau_{X_i})$

out of their product topological space

$\array{ X_1 \times X_2 &\overset{pr_1}{\longrightarrow}& X_1 \\ (x_1, x_2) &\overset{\phantom{AAA}}{\mapsto}& x_1 }$
$\array{ X_1 \times X_2 &\overset{pr_2}{\longrightarrow}& X_2 \\ (x_1, x_2) &\overset{\phantom{AAA}}{\mapsto}& x_2 }$

are open continuous functions (def. 1).

This is because, by definition, every open subset $O \subset X_1 \times X_2$ in the product space topology is a union of products of open subsets $U_i \in X_1$ and $V_i \in X_2$ in the factor spaces

$O = \underset{i \in I}{\cup} \left( U_i \times V_i \right)$

and because taking the image of a function preserves unions of subsets

\begin{aligned} pr_1\left( \underset{i \in I}{\cup} \left( U_i \times V_i \right) \right) & = \underset{i \in I}{\cup} pr_1 \left( U_i \times V_i \right) \\ & = \underset{i \in I}{\cup} U_i \end{aligned} \,.

Last revised on June 7, 2017 at 07:34:53. See the history of this page for a list of all contributions to it.