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.

Notes

Proofs: 1
Propositions: 2


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References

Bibliography

  1. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
  2. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition