Dense-in-itself

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

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 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 ${\displaystyle x}$ contains at least one other irrational number ${\displaystyle y\neq x}$. 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 ${\displaystyle \mathbb {R} }$. As an example that is dense-in-itself but not dense in its topological space, consider ${\displaystyle \mathbb {Q} \cap [0,1]}$. This set is not dense in ${\displaystyle \mathbb {R} }$ 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]
• In a topological space, the intersection of an open set and a dense-in-itself set is dense-in-itself.
• In a topological space, the closure of a dense-it-itself set is a perfect set.[6]

See also

• Nowhere dense set
• Glossary of topology
• Dense order

Notes

1. Steen & Seebach, p. 6
2. Engelking, p. 25
3. http://www.topo.auburn.edu/tp/reprints/v21/tp21008.pdf
4. https://www.researchgate.net/publication/228597275_a-Scattered_spaces_II
5. Engelking, 1.7.10, p. 59
6. 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.

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