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sequentially compact (Definition)

A topological space $ X$ is sequentially compact if every sequence in $ X$ has a convergent subsequence.

When $ X$ is a metric space, the following are equivalent:



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"sequentially compact" is owned by mps.
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See Also: compact, weakly countably compact, Bolzano-Weierstrass theorem

Other names:  sequential compactness
Keywords:  topology, sequence, convergence

Attachments:
relationship among different kinds of compactness (Theorem) by rm50
proof that a metric space is compact if and only if it is complete and totally bounded (Theorem) by rm50
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Cross-references: complete, totally bounded, compact, limit point compact, the following are equivalent, metric space, subsequence, convergent, sequence, topological space
There are 9 references to this entry.

This is version 3 of sequentially compact, born on 2002-07-06, modified 2004-02-09.
Object id is 3159, canonical name is SequentiallyCompact.
Accessed 10335 times total.

Classification:
AMS MSC: 54D30 (General topology :: Fairly general properties :: Compactness)
 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
Pending Errata and Addenda
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don't you need separability? by paolini on 2002-12-13 12:05:52
Are you sure that
 "Sequential compactness is equivalent to compactness when $X$ is a metric space."?

I can prove this claim only when $X$ is metric and separable.

Em.

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