Sequential compactness

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The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence.

Sequential compactness extends this notion to general topological spaces.

Definition

A topological space (X,J) is sequentially compact if every (infinite) Sequence has a convergent subsequence.

Common forms

Functional Analysis

A subset S of a normed vector space (V,,F) is sequentially compact if any sequence (an)n=1k has a convergent subsequence (ani)i=1, that is (ani)i=1aK

Like with compactness, we consider the subspace topology on a subset, then see if that is compact to define "compact subsets" - we do the same here. As warned below a topological space is not sufficient for sequentially compact compact, so one ought to use a metric subspace instead. Recalling that a norm can give rise to the metric d(x,y)=xy

Warning

Sequential compactness and compactness are not the same for a general topology

Uses

  • A metric space is compact if and only if it is sequentially compact, a theorem found here