Sequentially compact space

In mathematics, a topological space is sequentially compact if every sequence into the space has a convergent subsequence.

Examples and properties

1. A metrizable space is sequentially compact if and only if it is compact. However, in general there exist sequentially compact spaces which are not compact and compact spaces that are not sequentially compact.
2. The space of all real numbers with the standard topology is not sequentially compact; the sequence (s_n) = n for all natural numbers n is a sequence which has no convergent subsequence.
3. One form of the Bolzano-Weierstrass theorem states that every bounded sequence in R has a convergent subsequence.
4. If X is a subspace of R, then X is sequentially compact if and only if it is compact. This follows from the fact that R is metrizable.
5. From example 4, one can easily prove the Bolzano-Weierstrass theorem as follows: If (s_n) is a bounded sequence in R, it must be a subset of [−m, m] for some integer m (since it is bounded). Since [−m, m] is compact and therefore sequentially compact, (s_n) must have a convergent subsequence (whose limit lies in [−m, m]).

See also

• Limit point compact
• Compact space
• Limit point
• Sequence
• Bolzano-Weierstrass theorem

References

• James Munkres (1999). Topology (2nd edition ed.). Prentice Hall. ISBN 0-13-181629-2. 

This topology-related article is a stub. You can help Wikipedia by expanding it. v • d • e