In general topology, a subset of a topological space is said to be dense-in-itself or crowded 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 .
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).
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.
- The union of any family of dense-in-itself subsets of a space X is dense-in-itself.
- Every open subset of a dense-in-itself space is dense-in-itself.
- Every dense subset of a dense-in-itself T1 space is dense-in-itself. Note that this requires the space to be T1; for example in the space with the indiscrete topology, the set is dense, but is not dense-in-itself.
- In a topological space, the closure of a dense-it-itself set is a perfect set.
- Steen & Seebach, p. 6
- Engelking, p. 25
- Engelking, 1.7.10, p. 59
- Kuratowski, p. 78
- Kuratowski, p. 78
- Kuratowski, p. 77
- 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.