TITLE: Embeddings are precisely proper injective immersions.
QUESTION [21 upvotes]: We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective immersions? Or could someone supply a proof?

REPLY [7 votes]: The necessary material is contained in Appendix A of John Lee's Introduction to Smooth Manifolds (and elsewhere, I'm sure, but I like this source). When using somewhat ambiguous terms like embedding, it's always best to define them:

We will say that $f:X\rightarrow Y$ is a topological embedding if it is homeomorphism onto its image (note that $f$ is continuous and injective).
We will say that $f:X\rightarrow Y$ is a "smooth embedding" if it is a topological embedding and a smooth immersion (here, $X$ and $Y$ are smooth manifolds).

Most of this has to do purely with topology. Only in the end do we treat smooth stuff, by tacking on "immersion" to all of the existing, equivalent conditions.
Assertion 1. If $f:X\rightarrow Y$ is a topological embedding with closed image, then $f$ is closed and proper.
Proof. If $C\subset X$ is closed, then $f(C)$ is closed in $f(X)$ (since $f:X\rightarrow f(X)$ is a homeomorphism) and thus in $Y$ (since $f(X)$ is closed in $Y$).
If $K\subset Y$ is compact, then $K\cap f(X)$ is compact (since $f(X)$ is closed in $Y$) and thus $f^{-1}(K)$ is compact (since $f:X\rightarrow f(X)$ is a homeomorphism). $\blacksquare$
Assertion 2. If $f:X\rightarrow Y$ is continuous, injective and closed, then $f$ is a topological embedding with closed image (and thus proper).
Proof. Since $f$ is closed, the image $f(X)$ is certainly closed in $Y$. Moreover, the map $f:X\rightarrow f(X)$ is a closed, continuous bijection, i.e. a homeomorphism. $\blacksquare$
These first two assertions show that $f$ is a topological embedding with closed image if and only if $f$ is continuous, injective and closed. Moreover, any such $f$ is a proper map.
Assertion 3. If $Y$ is locally compact Hausdorff (or compactly generated Hausdorff) and $f:X\rightarrow Y$ is continuous and proper, then $Y$ is closed.
Proof. See this MSE post or Lee's Theorem A.57. $\blacksquare$
By Assertion 3, we see that if $Y$ is locally compact Hausdorff (e.g. a manifold) and $f:X\rightarrow Y$ is a continuous injection, then the following are equivalent:

$f$ is proper,
$f$ is closed,
$f$ is a topological embedding with closed image.

Thus if $X$ and $Y$ are smooth manifolds, the following are equivalent:

$f$ is a proper, injective immersion,
$f$ is a closed, injective immersion,
$f$ is a smooth embedding with closed image.