# sequentially compact

A topological space $X$ 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 $W(2\omega_{1})$ is sequentially compact but not first countable, since $\omega_{1}$ has not countable 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 $X=I^{I}$, where $I$ is the closed unit interval (with the usual topology), and $X$ is equipped with the product topology. Then $X$ is compact (since $I$ is, together with Tychonoff theorem). However, $X$ is not sequentially compact. To see this, let $f_{n}:I\to I$ be the function such that for any $r\in I$, $f(r)$ is the $n$-th digit of $r$ in its binary expansion. But the sequence $f_{1},\ldots,f_{n},\ldots$ has no convergent subsequences: if $f_{n_{1}},\ldots,f_{n_{k}},\ldots$ is a subsequence, let $r\in I$ such that its binary expansion has its $k$-th digit $0$ iff $k$ is odd, and $1$ otherwise. Then $f_{n_{1}}(r),\ldots,f_{n_{k}}(r),\ldots$ is the sequence $0,1,0,1,\ldots$, and is clearly not convergent. The ordinal space $\Omega_{0}:=W(\omega_{1})$ is an example of a sequentially compact space that is not compact, since the cover $\{W(\alpha)\mid\alpha\in\Omega_{0}\}$ has no finite subcover.

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

• $X$ is sequentially compact.

• $X$ is limit point compact.

• $X$ is compact.

• $X$ is totally bounded and complete.

 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