TITLE: Understanding the definition of "proper maps" in Differential Topology by Guillemin and Pollack
QUESTION [5 upvotes]: 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?

REPLY [4 votes]: 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.