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in itself

Dense-in-itself

In mathematics, 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.

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