Sequentially compact space

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 ${\displaystyle X}$.

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). However, there exist sequentially compact topological spaces that are not compact, and compact topological spaces that are not sequentially compact.