countably compact space

A countably compact space is always limit point compact.

Every countably compact space is pseudocompact.

Countably compact spaces are pseudocompact and weakly countably compact.

Not every countably compact space is compact; an example is given by the first uncountable ordinal with the order topology.

Every countably compact space (and hence every compact space) is weakly countably compact, but the converse is not true.

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.

Two classes of topological spaces are given Frolík's name: the class P of all spaces such that is pseudocompact for every pseudocompact space , and the class C of all spaces such that is countably compact for every countably compact space .