A topological space is said to be weakly countably compact (or limit point compact) if every infinite subset of has a limit point.

Every countably compact space is weakly countably compact. The converse is true in spaces.

A metric space is weakly countably compact if and only if it is compact.

An easy example of a space that is not weakly countably compact is any infinite set with the discrete topology. A more interesting example is the countable complement topology on an uncountable set.