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Definition: Derived, Dense-in-itself, and Perfect Sets
Let $(X,\mathcal O)$ be a topological space.
* The set $V$ of all limit points of a subset $U\subset X$ is called derived of $U$.
* If $U$ contains no isolated points, it is called dense-in-itself.
* A closed dense-in-itself set $U$ is called perfect.
- Generally, derived sets contain some points of $U$ and some points of the complement $U^C$.
- By definition, dense-in-itself and perfect sets contain no isolated points.
- But this is also the case for derived sets: Any isolated point of $U$ is necessarily not in a derived set of $U,$ otherwise there would be open sets in $U$ containing both, the isolated point and a distinct limit of $U.$
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- Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
- Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition