Compact set

The notion of Compact sets is very important in the field of topology

Definition

Let $S\subset X$

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

The set $S$ is said to be Compact if and only if for every $\{G_{\alpha}\}$ that is an open cover 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 $S$

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Compact set

Definition