Dense-in-itself

In mathematics, a subset ${\displaystyle A}$ of a topological space is said to be dense-in-itself if ${\displaystyle A}$ contains no isolated points.

Every dense-in-itself closed set is perfect. 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 (considered as a subset of the real numbers). This set is dense-in-itself because every neighborhood of an irrational number ${\displaystyle x}$ contains at least one other irrational number ${\displaystyle y\neq x}$. 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 (also considered as a subset of the real numbers) is also dense-in-itself but not closed.

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