Is the notion of a proper map useful between non-locally-compact spaces?

Recall that a map $f: X \to Y$ between topological spaces is called proper if, for every compact $K \subseteq Y$, $f^{-1}(K)$ is compact.

It strikes me that this definition is unlikely to be useful if $Y$ doesn't have "enough" compact subsets. And it's likely to be "too restrictive" if $X$ doesn't have "enough" compact subsets.

This is born out in the intuitive picture (cf. wikipedia) which says that if $f: X \to Y$ is proper, and $\{x_i\}$ is a sequence which "escapes to infinity" in the sense that any compact $K \subseteq X$ contains at most finitely many of the $x_i$'s, then the sequence $\{f(x_i)\}$ escapes to infinity in the same sense. This notion could be modified to use nets instead of sequences, but it would still be the wrong notion of "escape to infinity" in a non-locally-compact space -- for example, in this sense an orthonormal basis of $\ell_2(\mathbb{N})$ "escapes to infinity". So this intuitive picture really only works for locally compact spaces, where it essentially says that $f$ extends to a map between 1-point compactifications sending $\infty$ to $\infty$.

Hence the question: is the notion of a proper map useful when one is working with non-locally-compact spaces? For example, are there any interesting theorems whose hypotheses ask that a map be proper without asking that the spaces involved be locally compact? If not, is there some sort of "substitute" notion which does work well for spaces that are not locally compact? (It would be nice, but not necessary, for such a substitute notion to agree with properness on locally compact spaces.)

Here's an example, although I'm not sure how useful it is (all of the common applications are for $X$ and $Y$ locally compact).

Theorem. Let $X$ and $Y$ be completely regular spaces. If $f:X\to Y$ is a continuous closed surjection and every $f^{-1}\{y\}$ is connected and compact (i.e., $f$ is perfect and monotone), then $\beta f:\beta X\to \beta Y$ is monotone.

It is a fairly deep result, and can be used to get strange continua in $\beta X\setminus X$.

An application where $X$ and $Y$ are nowhere locally compact: Let $X=\mathbb Q \times [0,1]$, $Y=\mathbb Q$, and let $f$ be the first coordinate projection. If $p\in \beta \mathbb Q$ then $\beta f^{-1}\{p\}$ is a continuum. Of course this is only interesting if $p\in \beta \mathbb Q\setminus \mathbb Q$, otherwise you just get the interval. I have no idea what properties these continua have, but it seems like they should be pretty odd.

I'm not sure this is exactly what you were looking for because the assumption regarding compact preimages is weaker than "proper."

• Thanks, this is neat! I guess the fact that the hypothesis is weakened to talk just about points rather than compact sets speaks to the fact that when you don't have "enough" compact sets, it's not very useful to talk about them. – tcamps Apr 17 '16 at 21:30
• Yes, for instance in $\mathbb Q$ the only compact sets other than singletons are finite unions of convergent sequences, I think. – Forever Mozart Apr 17 '16 at 22:06

I just recalled a fact that uses properness but not local compactness. Namely, if $X$ is Hausdorff, then a local homeomorphism $f: X \to Y$ is proper iff it is a covering map iff it is a finite cover. But again, properness can be weakened here to the condition the the preimage of any point is finite.