We understand the one-point compactification of a topological space $X$ is the special way to build a compact space from $X$ by adjoining just one additional point such that $X$ is densely embedded.
I am looking for a few suggestions:
(a) What are the tricky questions one could expect?
(b) A good motivation to study such compactification.
(c) A few good applications of one-point compactification.
Thank you in advance. Any help will be appreciated.
Let $f: X \to Y$ be a proper continuous map where $Y$ is locally compact Hausdorff. Then $f$ is closed. This is a sort of variant of the tube lemma and is useful in application.
You can prove this by hand (your proof will basically boil down to showing that $Y$ is compactly generated, and reduce to checking this over compact sets where the result is standard). But the following trick makes it easier.
If $Y$ is locally compact Hausdorff then $Y*$ is Hausdorff.
Then $f*: X* \to Y*$ is continuous from compact to Hausdorff hence closed. Now given closed $C$ in $X$, we have that $f*(C*) = f(C)*$ is closed, from which we conclude that $f(C)$ is closed in $X$.