Hausdorff space into a compact Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space.

Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed.

Closed sets also give a useful characterization of compactness: a topological space is compact if and only if every collection of nonempty closed subsets of with empty intersection admits a finite subcollection with empty intersection.

A topological space is disconnected

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