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The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness, countable compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Čech) and other topics closely related to compactness.

Compactness is a topological property. We say that a topological space $X$ is compact if whenever we cover $X$ by a collection of open sets we can find a finite number of open sets from the collection which cover $X$. For example, $[0,1]$ is a compact subspace of $\mathbb{R}$, but $(0,1)$ and $\mathbb{R}$ are not.

We say that a space $X$ is sequentially compact if every sequence has a convergent subsequence. These properties are equivalent for metric space, although neither implies the other in general.

More information can be found on Wikipedia.

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