I saw that in Mukresh's book there is an example which shows that limit point compactness may not imply compactness.
The example says $Z_+ \times Y$ is the space where every subset has limit point. Where $Y$ has two points only and trivial topology has been taken.
Would not $Z \times Y$ be limit point compact set but not compact set?
Best Answer
$Z \times Y$ is not compact: consider the open cover $\{n\} \times Y$, $n \in Z$, of which we cannot omit a member or we have no cover.
It is limit point compact because if $A$ is any subset of the space and $(n,i) \in A$, $(n,i')$ is a limit point of $A$ (where $i'$ is the other point than $i$ in the two point set $Y$).
Taking $Z$ or $Z^+$ is the same thing here.