- 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. This set is dense-in-itself because every neighborhood of an irrational number x 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.
See also
This article incorporates material from Dense in-itself on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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