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单词 SequentiallyCompact
释义

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:

  • X is sequentially compact.

  • X is limit point compact.

  • X is compact.

  • X is totally boundedPlanetmathPlanetmath and completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Titlesequentially compact
Canonical nameSequentiallyCompact
Date of creation2013-03-22 12:50:05
Last modified on2013-03-22 12:50:05
Ownermps (409)
Last modified bymps (409)
Numerical id12
Authormps (409)
Entry typeDefinition
Classificationmsc 40A05
Classificationmsc 54D30
Synonymsequential compactness
Related topicCompact
Related topicLimitPointCompact
Related topicBolzanoWeierstrassTheorem
Related topicNet
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更新时间:2025/5/4 11:30:48