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I know that if $X$ is a Hausdorff topological space and $A$ is compact in $X$, then $A$ is closed in $X$. My question is that if $A$ is a closed set in $X$ (where $X$ is Hausdorff), what extra condition is needed to ensure that $A$ is compact in $X$?

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2 Answers 2

up vote 4 down vote accepted

There’s really not much that can be said in general: it depends very much on the Hausdorff space $X$. Some examples:

  • If $X$ is compact, then every closed subset of $X$ is compact.

  • If $X$ is $\mathbb{R}^n$ for some $n$, a closed subset of $X$ is compact if and only if it’s bounded in the usual metric.

  • If $X$ is an infinite space with the discrete topology, a closed subset of $X$ is compact if and only if it’s finite.

  • If $X$ is the space $\omega_1$ of countable ordinals with the order topology, a closed subset of $X$ is compact if and only if it’s countable.

  • If $X$ is the Sorgenfrey line, every compact subset of $X$ is countable, but $X$ has closed subsets, even bounded ones, that are not compact, e.g., $A=\{1\}\cup\{1-2^{-n}:n\in\omega\}$: $\{[1-2^{-n},1-2^{1-n}):n\in\omega\}\cup\{[1,2)\}$ is an open cover of $A$ with no finite subcover.

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@Jonas: No, I meant discrete; I just lost track of what I was doing. The cofinite topology isn’t Hausdorff, so it’s excluded from consideration anyway. But thanks for catching the silly error. –  Brian M. Scott Dec 6 '11 at 2:17
    
Can I know what is $\omega$ when you write $n \in \omega$? –  user38268 Jul 12 '12 at 11:50
    
@BenjaLim: The first infinite ordinal, which is the set of all finite ordinals. I could just as well have written $n\in\Bbb N$, where $\Bbb N=\{0,1,2,\dots\}$. –  Brian M. Scott Jul 12 '12 at 19:14

Brian has already given you many examples explaining that compactness of $A$ strongly depends on properties of the Hausdorff space $X$. It is worth to add to his list the following one:

Let $(X,\rho)$ be a metric space (thus Hausdorff) and $A\subseteq X$. Then $A$ is compact if and only if $A$ is complete and totally bounded.

This is one of the most important characterisations of compact subsets of metric spaces.

Many interesting properties and other characterisations you can find in the Wikipedia article: http://en.wikipedia.org/wiki/Compact_space

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