dense in-itself
A subset of a topological space is said to be dense-in-itself if contains no isolated points
.
Note that if the subset is also a closed set, then 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 contains at least one other irrational number . 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.