In general topology, a subset of a topological space is said to be dense-in-itself[1][2] or crowded[3][4] if has no isolated point. Equivalently, is dense-in-itself if every point of is a limit point of . Thus is dense-in-itself if and only if , where is the derived set of .
A dense-in-itself closed set is called a perfect set. (In other words, a perfect set is a closed set without isolated point.)
The notion of dense set is unrelated to dense-in-itself. This can sometimes be confusing, as "X is dense in X" (always true) is not the same as "X is dense-in-itself" (no isolated point).
Examples
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 contains at least one other irrational number . On the other hand, the 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 . As an example that is dense-in-itself but not dense in its topological space, consider . This set is not dense in but is dense-in-itself.
![{\mathbb {Q}}\cap [0,1]](http://img.tfd.com/wiki/math/70/822d49c61001a9fd0fee4578855b367df40dc4ca.svg)
Properties
- The union of any family of dense-in-itself subsets of a space X is dense-in-itself.[5]
- In a topological space, the intersection of an open set and a dense-in-itself set is dense-in-itself.
- In a topological space, the closure of a dense-it-itself set is a perfect set.[6]
See also
- Nowhere dense set
- Glossary of topology
- Dense order
Notes
- ^ Steen & Seebach, p. 6
- ^ Engelking, p. 25
- ^ http://www.topo.auburn.edu/tp/reprints/v21/tp21008.pdf
- ^ https://www.researchgate.net/publication/228597275_a-Scattered_spaces_II
- ^ Engelking, 1.7.10, p. 59
- ^ Kuratowski, p. 77
References
- Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4.
- Kuratowski, K. (1966). Topology Vol. I. Academic Press. ISBN 012429202X.
- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446.