Compact set

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Compactness is a topological property that appears in a wide variety of contexts. In particular, it is a "tameness property" that tells you that the objects you are dealing with are in some sense well-behaved.

Definition

Let $X$ be a topological space and let $S\subset X$ .

A set of open sets $G_{\alpha}\subset X$ is said to be an open cover of $S$ if $S\subset\bigcup_{\alpha}G_{\alpha}$ .

The set $S$ is said to be compact if and only if for every open cover $\{G_{\alpha}\}$ of $S$ , there exists a finite set $\{\alpha_1,\alpha_2,\ldots,\alpha_n\}$ such that $\{G_{\alpha_k}\}_{k=1}^{n}$ is also an open cover of $S$ . This is often expressed in the sentence, "A set is compact if and only if every open cover admits a finite subcover."

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Compact set

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Definition