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.

## Properties

- The union of any family of dense-in-itself subsets of a space X is dense-in-itself.
^{[5]} - Every open subset of a dense-in-itself space is dense-in-itself.
^{[6]} - Every dense subset of a dense-in-itself T
_{1}space is dense-in-itself.^{[7]}Note that this requires the space to be T_{1}; 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.
^{[8]}

## See also

## 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. 78**^**Kuratowski, p. 78**^**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.

*This article incorporates material from Dense in-itself on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*