Compact space

A topological space is compact if every cover by open sets has a finite subcover. Compactness generalizes the finite-cover behavior of closed bounded subsets of Euclidean space.

Open cover: A family of open sets whose union contains a given space or subset. Compactness asks whether every such cover can be reduced to finitely many sets.
Heine–Borel theorem: A theorem stating that a subset of Euclidean space is compact exactly when it is closed and bounded. It translates open-cover compactness into a familiar geometric test in Euclidean space.
Arzelà–Ascoli theorem: A theorem giving conditions under which a family of continuous functions has compact closure in the uniform topology. It uses compactness to extract convergent subsequences of functions.
Local compactness: A space is locally compact when each point has a neighborhood with compact closure, or an equivalent compact-neighborhood condition. It is a local condition and does not require the whole space to be compact.
Subcover: A subfamily of a cover that still covers the same space or subset. The defining condition requires a finite subcover from every open cover.
Extreme value theorem: A theorem stating that a continuous real-valued function on a compact space attains a maximum and minimum. Compactness ensures continuous functions cannot escape their extrema.
Tychonoff theorem: A theorem stating that any product of compact topological spaces is compact in the product topology. It makes compactness available in products with arbitrarily many factors.
Countable compactness: A space is countably compact if every countable open cover has a finite subcover. It restricts the cover condition to countable covers, making it weaker in general.
Topological space: A set equipped with a collection of subsets called open sets, satisfying axioms for unions and finite intersections. Compactness is defined using the open sets of this underlying structure.
Finite intersection property: A family of sets has this property when every finite subfamily has nonempty intersection. Compactness is equivalent to a finite-intersection condition for closed sets.

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