Understanding the definition of "proper maps" in Differential Topology by Guillemin and Pollack

Question

I am trying to understand a piece of definition in Differential Topology by Gullemin and Pllack (GP). Before introducing the concept of "embedding", GP gives the following definition (page 17):

A map $f:X\to Y$ is called proper if the peimage of every compact set in $Y$ is compact in $X$. Intuitively, a proper map is one that maps points "near infinity" in $X$ to points "near infinity" in $Y$.

I do not understand the "intuition" part: what does it really mean (mathematically) and how does it relate to the definition of proper maps?

Answer 1

Compact sets are bounded in metric spaces. Assuming that our intuition of a compact set in a topological space is something that's not very large and we have some sort of control over its size compared to our space, the definition of a proper map says that the preimage of a compact set, i.e. something that is not very large, must be something that is not very large as well.

In a contrapositive sense, intuitively, that is like saying that if something is very large that its size gets out of hand, i.e. large enough to get close to infinity in some sense, then it must be mapped to something that is very large as well, i.e. "near infinity".

Answer 2

Let $f:X\to Y$ be a continuous function between Hausdorff topological spaces. Let us consider the one-point Alexandroff compactifications $\overline{X}=X\cup\{\infty\}$ and $\overline{Y}=Y\cup\{\infty\}$. Then $f$ extends to a continuous function $\overline{f}:\overline{X}\to \overline{Y}$ such that $\overline{f}(\infty)=\infty$ if and only if $f$ is proper.