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 distinct from 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 that is dense-in-itself but not closed (and hence not a perfect set) is the set 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. Similarly, the set of rational numbers is also dense-in-itself but not closed in the space of real numbers.

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

A singleton subset of a space can never be dense-in-itself, because its unique point is isolated in it.

The dense-in-itself subsets of any space are closed under unions.[5] In a dense-in-itself space, they include all open sets.[6] In a dense-in-itself T1 space they include all dense sets.[7] However, spaces that are not T1 may have dense subsets that are not dense-in-itself: for example in the dense-in-itself space with the indiscrete topology, the set is dense, but is not dense-in-itself.

The closure of any dense-in-itself set is a perfect set.[8]

In general, the intersection of two dense-in-itself sets is not dense-in-itself. But the intersection of a dense-in-itself set and an open set is dense-in-itself.

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.

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