Dense-in-itself explained

In general topology, a subset

A of a topological space is said to be dense-in-itself^{[1] [2]} or crowded^{[3] [4]} if A has no isolated point.Equivalently, A is dense-in-itself if every point of A is a limit point of A .Thus A is dense-in-itself if and only if A\subseteq A' , where A' is the derived set of A .

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

x contains at least one other irrational number y ≠ x . 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

R . As an example that is dense-in-itself but not dense in its topological space, consider Q \cap [ 0 , 1 ] . This set is not dense in R but is dense-in-itself.

Properties

A singleton subset of a space

X 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

X = \{a,b\} with the indiscrete topology, the set A = \{a\} 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

• Nowhere dense set
• Glossary of topology
• Dense order

References

• Book: Engelking, Ryszard. Ryszard Engelking. General Topology. Heldermann Verlag, Berlin. 1989. 3-88538-006-4.
• Book: Kuratowski, K.. Kuratowski. Academic Press . 1966. Topology Vol. I. 012429202X.
• Book: Steen . Lynn Arthur . Lynn Arthur Steen . Seebach . J. Arthur Jr. . J. Arthur Seebach, Jr. . . 1978 . . Berlin, New York . Dover reprint of 1978 . 978-0-486-68735-3 . 507446.

Notes and References

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