Difference between revisions of "Sequential compactness"
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Revision as of 22:26, 8 March 2015
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.
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
