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Dense-in-itself

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 $y\neq x$ . 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 $\mathbb {R}$ . As an example that is dense-in-itself but not dense in its topological space, consider $\mathbb {Q} \cap [0,1]$ . This set is not dense in $\mathbb {R}$ 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.

Notes

^ Steen & Seebach, p. 6
^ Engelking, p. 25
^ Levy, Ronnie; Porter, Jack (1996). "On Two questions of Arhangel'skii and Collins regarding submaximal spaces" (PDF). Topology Proceedings. 21: 143–154.
^ Dontchev, Julian; Ganster, Maximilian; Rose, David (1977). "α-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.

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Dense-in-itself

Examples

Properties

See also

Notes

References