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In a Hausdorff space, every compact set is closed
Compactness implies closedness: Let be a Hausdorff space and let be compact. Then is closed in .
In particular, this applies to subsets of any metric space, since metric spaces are Hausdorff.
A Hausdorff space (or T2 space) is a topological space such that for any distinct points there exist neighborhoods of and of with .
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
A closed set in a topological space is a subset whose complement is a open set.