Dense-in-itself

In general topology, a subset $A$ of a topological space is said to be dense-in-itself or crowded
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 $\mathbb$. As an example that is dense-in-itself but not dense in its topological space, consider \mathbb \cap ,1/math>. This set is not dense in $\mathbb$ 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. In a dense-in-itself space, they include all open sets. In a dense-in-itself T₁ space they include all dense sets. 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=\$ with the indiscrete topology, the set $A=\$ is dense, but is not dense-in-itself.

The closure of any dense-in-itself set is a perfect set.Kuratowski, p. 77

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

References

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{{PlanetMath attribution, id=6228, title=Dense in-itself

Topology