Sequentially compact space
- Sequentially compact space
In mathematics, a topological space is sequentially compact if every sequence in 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 compact spaces which are not sequentially compact, and conversely.
2. The space of all real numbers with the standard topology is not sequentially compact; the sequence (sn) = 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 compact subspace of R, then "X" is sequentially 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 (sn) is a bounded sequence in R, it must be a subset of [-n, n] for some integer n (since it is bounded). Since [-n, n] is compact and therefore sequentially compact, (sn) must have a convergent subsequence (whose limit lies in [-n, n] ).
ee also
* Limit point compact
* Compact space
* Limit point
* Sequence
* Bolzano-Weierstrass theorem
References
* cite book
author = James Munkres
year = 1999
title = Topology
edition = 2nd edition
publisher = Prentice Hall
id = ISBN 0-13-181629-2
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