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]

## 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