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Compactness

Definitions

Given a topological space $(X, \mathbb{T})$, an (open) cover is a set $\mathcal{U}$ of open sets in $X$, whose union is the whole of $X$. A subcover is a subset of a cover that is also a cover.

If every cover has a finite subcover, we say $X$ is compact.

An open cover for a subset $A \subset X$ is a set of open sets in $X$ whose union contains the whole of $A$.

A subset is compact if it is compact as a subspace, or alternatively if every open cover has a finite subcover.

Properties

Any compact subset of a Hausdorff space is closed.

Examples

Singletons are compact.

all these pages adapted with probably insufficient credit from my university's lecture notes