sequentially compact space

As a consequence of the above result, every sequentially compact space is pseudocompact.

Sequentially compact spaces are countably compact.

An only slightly more elaborate "diagonalization" argument establishes the sequential compactness of a countable product of sequentially compact spaces.

Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact.

It is almost trivial to prove that the product of two sequentially compact spaces is sequentially compact - one passes to a subsequence for the first component and then a subsubsequence for the second component.

This is known as sequentially compact space and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countably compact space and compact space defined via open covers.

However in general there exist sequentially compact spaces which are not compact (such as the first uncountable ordinal with the order topology), and compact spaces which are not sequentially compact (such as the product of uncountably many copies of the closed unit interval).