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

See also

Notes

  1. ^ Steen & Seebach, p. 6
  2. ^ Engelking, p. 25
  3. ^ Levy, Ronnie; Porter, Jack (1996). "On Two questions of Arhangel'skii and Collins regarding submaximal spaces" (PDF). Topology Proceedings. 21: 143–154.
  4. ^ Dontchev, Julian; Ganster, Maximilian; Rose, David (1977). "α-Scattered spaces II".
  5. ^ Engelking, 1.7.10, p. 59
  6. ^ Kuratowski, p. 78
  7. ^ Kuratowski, p. 78
  8. ^ Kuratowski, p. 77

References

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