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Wikipedia / Sequentially compact space

In mathematics, a topological space is sequentially compact if every sequence has a convergent subsequence.

Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (using the axiom of countable choice). However, there exist sequentially compact topological spaces that are not compact, and compact topological spaces that are not sequentially compact.

Examples and properties

The space of all real numbers with the standard topology for all natural numbers is a sequence that has no convergent subsequence.

On a first countable space has a convergent subsequence if and only if

be a countable neighborhood base is a well-ordered set.

A point in the above intersection is called a cluster point. Thus, for first countable spaces, the definition of a sequentially compact space is the same as saying that each sequence in the space has a cluster point.

If a space is a metric space, then it is sequentially compact if and only if it is compact (cf. ). Here is how to see this, using only the countable Choice such that

of nonempty closed subsets has nonempty intersection. But this is clear since

The first uncountable ordinal with the order topology is an example of a sequentially compact topological space that is not compact. The topological product copies of the closed unit interval is an example of a compact space that is not sequentially compact.

Related notions

A topological space is said to be limit point compact if every infinite subset of has a limit point in , and countably compact if every countable open cover has a finite subcover. In a metric space, the notions of sequential compactness, limit point compactness, countable compactness and compactness are all equivalent (if one assumes the axiom of choice).

In a sequential (Hausdorff) space sequential compactness is equivalent to countable compactness.

There is also a notion of a one-point sequential compactification—the idea is that the non convergent sequences should all converge to the extra point.

See also

Notes

References