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

Every countably compact space is weakly countably compact. The converse is true in $\mathrm{T}_1$ spaces.

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

An easy example of a space $X$ 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.