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

Statement

Proposition

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

Let

1. $X$ be a topological space

2. $Y$ a locally compact topological space

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

Then the following are equivalent

1. $f$ is an injective proper map,

2. $f$ is a closed embedding of topological spaces.

Proof

In one direction, if $f$ is an injective proper map, then since proper maps to locally compact spaces are closed, it follows that $f$ is also closed map. The claim then follows since closed injections are embeddings, and since the image of a closed map is closed.

Conversely, if $f$ is a closed embedding, we only need to show that the embedding map is proper. So for $C \subset Y$ a compact subspace, we need to show that the pre-image $f^{-1}(C) \subset X$ is also compact. But since $f$ is an injection (being an embedding), that pre-image is just the intersection $f^{-1}(C) \simeq C \cap f(X)$. By the nature of the subspace topology, this is compact if $C$ is.

Related statements

• closed injections are embeddings

• proper maps to locally compact spaces are closed