In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in
Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if one assumes countable choice).
The space of all real numbers with the standard topology is not sequentially compact; the sequence
[1] The first uncountable ordinal with the order topology is an example of a sequentially compact topological space that is not compact.
copies of the closed unit interval is an example of a compact space that is not sequentially compact.
is said to be limit point compact if every infinite subset of
In a sequential (Hausdorff) space sequential compactness is equivalent to countable compactness.
[3] 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.