# 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 [ilmath](X,\mathcal{J})[/ilmath] is sequentially compact if every (infinite) Sequence has a convergent subsequence.

### Common forms

#### Functional Analysis

A subset [ilmath]S[/ilmath] of a normed vector space $(V,\|\cdot\|,F)$ is sequentially compact if any sequence $(a_n)^\infty_{n=1}\subset k$ has a convergent subsequence $(a_{n_i})_{i=1}^\infty$, that is $(a_{n_i})_{i=1}^\infty\rightarrow a\in K$

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 $\iff$ 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