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 $f_1,\mathrm{\dots },f_n,\mathrm{\dots }$ has no convergent subsequences: if $f_{n_1},\mathrm{\dots },f_{n_k},\mathrm{\dots }$ 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 $f_{n_1}(r),\mathrm{\dots },f_{n_k}(r),\mathrm{\dots }$ is the sequence $0,1,0,1,\mathrm{\dots }$, and is clearly not convergent. The ordinal space ${\mathrm{\Omega }}_0:=W({\omega }_1)$ is an example of a sequentially compact space that is not compact, since the cover $\{W(\alpha )\mid \alpha \in {\mathrm{\Omega }}_0\}$ 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