Dense-in-itself

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In mathematics, a subset A of a topological space is said to be dense-in-itself if 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 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 (also considered as a subset of the real numbers) is also dense-in-itself but not closed.

The above examples, the irrationals and the rationals, are also dense sets in their topological space, namely \mathbb{R}. As an example that is dense-in-itself but not dense in its topological space, consider \mathbb{Q} \cap [0,1]. This set is not dense in \mathbb{R} but is dense-in-itself.

It is also interesting to note, although tautological, that the domain of a continuous function must be the union of dense-in-itself sets and/or isolated points.[citation needed]

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This article incorporates material from Dense in-itself on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.