A subset $A$ of a topological space is said to be dense-in-itself if $A$ contains no isolated points.

Note that if the subset $A$ is also a closed set, then $A$ will be a perfect set. Conversely, every perfect set is dense-in-itself.

A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers. This set is dense-in-itself because every neighborhood of an irrational number $x$ contains at least one other irrational number $y\neq x$. On the other hand, this set of irrationals is not closed because every rational number lies in its closure.

For similar reasons, the set of rational numbers is also dense-in-itself but not closed.