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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.

Surojit
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    Well, certainly compactness is a good property and the one-point compactification is the smallest space containing $X$ that is compact. One also immediately sees folklore examples like projective spaces (in complex analysis also the Riemann sphere). – Qi Zhu Oct 02 '20 at 08:12
  • @QiZhu: Thank you for your comment. I am interested in why compactness is a good property ( a reason to motivate undergrad students). Also, some tricky questions one should care about. – Surojit Oct 02 '20 at 09:30
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    One tricky question is: when $X^{*}$ is Hausdorff? The answer is: if and only if $X$ is Hausdorff and locally compact. – freakish Oct 02 '20 at 09:35
  • See e.g. https://math.stackexchange.com/questions/485822/why-is-compactness-so-important/485856. I'm not sure what a tricky question is supposed to be, just look up any property of one-point compactifications and try to prove it. – Qi Zhu Oct 02 '20 at 09:40
  • Here is a nice application: If $X$ is locally compact, then it's rather easy to see, that it embedds into its one-point compactification. Compact spaces are completely regular, hence $X$ is completely regular. – Ulli Oct 03 '20 at 10:12

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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 $X$.

just simple things
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