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]
• Every open subset of a dense-in-itself space is dense-in-itself.^[6]
• Every dense subset of a dense-in-itself T₁ space is dense-in-itself.^[7] Note that this requires the space to be T₁; for example in the space ${\displaystyle X=\{a,b\}}$ with the indiscrete topology, the set ${\displaystyle A=\{a\}}$ is dense, but is not dense-in-itself.
• A singleton subset of a space ${\displaystyle X}$ is never dense-in-itself (because its unique point is isolated in it).
• In a topological space, the closure of a dense-it-itself set is a perfect set.^[8]

See also[]

• Nowhere dense set
• Glossary of topology
• Dense order

Notes[]

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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