nLab proper map

Contents

This entry is about the concept in topology. For variants see at proper morphism.

Contents

Definition

Definition

(proper maps)

A continuous function $f \colon (X, \tau_X) \to (Y, \tau_Y)$ is called proper if for $C \in Y$ a compact topological subspace of $Y$, then also its pre-image $f^{-1}(C)$ is compact in $X$.

Properties

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 proper map (def. ))

Definition

(locally compact topological space)

A topological space $X$ is called locally compact if for every point $x \in X$ and every open neighbourhood $U_x \supset \{x\}$ there exists a smaller open neighbourhood $V_x \subset U_x$ whose topological closure is compact (def. ) and still contained in $U$:

$\{x\} \subset V_x \subset \underset{\text{compact}}{Cl(V_x)} \subset U_x \,.$
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 (def. ), then it is a closed map.

Last revised on April 2, 2019 at 02:03:25. See the history of this page for a list of all contributions to it.