Sequential compactness
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=1⊂k has a convergent subsequence (ani)∞i=1, that is (ani)∞i=1→a∈K
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)=∥x−y∥
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