nLab closed injections are embeddings

Statement

Proposition

(open/closed continuous injections are embeddings)

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

1. an injective function

2. an open map or a closed map

is a topological embedding.

Proof

If $f$ is injective, then the map onto its image $X \to f(X) \subset Y$ is a bijection. Moreover, it is still continuous with respect to the subspace topology on $f(X)$. Now a bijective continuous function is a homeomorphism precisely if it is an open map or a closed map (by this prop.). But the image projection of $f$ has this property, respectively, if $f$ does (by this prop.).

Related concepts

• proper maps to locally compact spaces are closed

• 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