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In a Hausdorff space, every compact subset is closed.
Compact subset of a Hausdorff space is closed: Let be a Hausdorff space. If is compact, then is a closed set in .
This is one of the basic structural features of Hausdorff spaces and is used repeatedly in compactness arguments, including the compact-to-Hausdorff homeomorphism criterion and uniqueness phenomena such as uniqueness of limits.
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 such that its complement (the set difference) is a open set.
Closed sets are the natural targets of the closure operation: the closure of is the smallest closed set containing . Closed sets also provide an equivalent formulation of continuity via preimages of closed sets.
Compact-to-Hausdorff homeomorphism criterion: Let be a continuous map that is bijective. If is compact and is Hausdorff, then is a homeomorphism (equivalently, is continuous).
Uniqueness of limits in Hausdorff spaces: Let be a Hausdorff space and let be a sequence in . If converges to both and , then .
This property is a key reason Hausdorff spaces behave like metric spaces with respect to convergence, and it is closely aligned with results like compact subsets are closed.