nLab proper maps to locally compact spaces are closed

Background

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 and still contained in $U$:

$$\{x\} \subset V_x \subset \underset{\text{compact}}{Cl(V_x)} \subset U_x \,.$$

Statement and proof

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.

Proof

Let $C \subset X$ be a closed subset. We need to show that every $y \in Y \backslash f(C)$ has an open neighbourhood $U_y \supset \{y\}$ not intersecting $f(C)$ (by this prop.).

By local compactness of $(Y,\tau_Y)$ (def. ), $y$ has an open neighbourhood $V_y$ whose topological closure $Cl(V_y)$ is compact. Hence since $f$ is proper, also $f^{-1}(Cl(V_y)) \subset X$ is compact. Then also the intersection $C \cap f^{-1}(Cl(V_y))$ is compact, and hence so is

$$f(C \cap f^{-1}(Cl(V_y))) = f(C) \cap (Cl(V)) \; \subset Y \,.$$

This is also a closed subset, since compact subspaces of Hausdorff spaces are closed. Therefore

$$U_y \coloneqq V_y \backslash ( f(C) \cap (Cl(V_y)) ) = V_y \backslash f(C)$$

is an open neighbourhod of $y$ not intersecting $f(C)$.

Related statements

• injective proper maps to locally compact spaces are equivalently the closed embeddings

• closed injections are embeddings

• maps from compact spaces to Hausdorff spaces are closed and proper

• CW-complexes are Hausdorff spaces

• Hausdorff spaces are sober, schemes are sober

• continuous image of a compact space is compact

• closed subsets of compact spaces are compact

• compact subspaces of Hausdorff spaces are closed

• closed subspaces of compact Hausdorff spaces are equivalently compact subspaces

• quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff

• paracompact Hausdorff spaces are normal

• paracompact Hausdorff spaces equivalently admit subordinate partitions of unity