sequentially compact

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

Every sequentially compact space is countably compact. Conversely, every first countable countably compact space is sequentially compact. The ordinal space W(2ω1) is sequentially compact but not first countable, since ω1 has not countableMathworldPlanetmath local basis.

Next, compactness and sequential compactness are not compatible. In other words, neither one implies the other. Here’s an example of a compact space that is not sequentially compact. Let X=II, where I is the closed unit interval (with the usual topology), and X is equipped with the product topology. Then X is compactPlanetmathPlanetmath (since I is, together with Tychonoff theoremMathworldPlanetmath). However, X is not sequentially compact. To see this, let fn:II be the function such that for any rI, f(r) is the n-th digit of r in its binary expansion. But the sequence f1,,fn, has no convergent subsequences: if fn1,,fnk, is a subsequence, let rI such that its binary expansion has its k-th digit 0 iff k is odd, and 1 otherwise. Then fn1(r),,fnk(r), is the sequence 0,1,0,1,, and is clearly not convergent. The ordinal space Ω0:=W(ω1) is an example of a sequentially compact space that is not compact, since the cover {W(α)αΩ0} has no finite subcover.

When X is a metric space, the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

Title sequentially compact
Canonical name SequentiallyCompact
Date of creation 2013-03-22 12:50:05
Last modified on 2013-03-22 12:50:05
Owner mps (409)
Last modified by mps (409)
Numerical id 12
Author mps (409)
Entry type Definition
Classification msc 40A05
Classification msc 54D30
Synonym sequential compactness
Related topic Compact
Related topic LimitPointCompact
Related topic BolzanoWeierstrassTheorem
Related topic Net