Sequential compactness

A topological space is sequentially compact if every sequence in it has a subsequence converging to a point in the space.

Convergent sequence: A sequence whose terms approach a limit in a specified topological space. Sequential compactness guarantees a convergent subsequence from every sequence.
Compact space: A topological space in which every open cover has a finite subcover. Compactness is the cover-based property often compared with sequential compactness.
Topology: A structure specifying which subsets of a set are open and thereby defining continuity and convergence. Convergence, and therefore sequential compactness, is determined by the topology.
Continuous image: The image of a set under a continuous function, with the topology inherited from the codomain. A continuous image of a sequentially compact space is sequentially compact.
Sequential compactness of products: The study of when a product of sequentially compact spaces remains sequentially compact. Unlike compactness, sequential compactness is not preserved by arbitrary products.
Subsequence: A sequence formed by selecting terms at strictly increasing indices from another sequence. The compactness condition asks for a convergent selection of this kind.
Countable compactness: A space in which every countable open cover has a finite subcover. It is a cover condition weaker than compactness and not generally equivalent to sequential compactness.
Topological space: A set equipped with a collection of open subsets satisfying the axioms of a topology. Sequential compactness is a property of topological spaces.
Uniform continuity: Continuity with a single input-distance bound that works uniformly across the domain. Continuous functions on sequentially compact metric spaces are uniformly continuous.
Sequential fan: A countable topological space whose neighborhood structure illustrates subtleties of sequential convergence. Sequential fans provide examples where sequence-based properties diverge from compactness.

Linked from 9 pages