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 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 ${\displaystyle x}$ 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. 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 ${\displaystyle \mathbb {R} }$. 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

A singleton subset of a space ${\displaystyle X}$ 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 ${\displaystyle X=\{a,b\}}$ with the indiscrete topology, the set ${\displaystyle A=\{a\}}$ 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

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

• 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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