Dense-in-itself

Short description: Topological subset with no isolated point

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

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 [math]\displaystyle{ x }[/math] contains at least one other irrational number [math]\displaystyle{ y \neq x }[/math]. 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 [math]\displaystyle{ \mathbb{R} }[/math]. As an example that is dense-in-itself but not dense in its topological space, consider [math]\displaystyle{ \mathbb{Q} \cap [0,1] }[/math]. This set is not dense in [math]\displaystyle{ \mathbb{R} }[/math] but is dense-in-itself.

Properties

A singleton subset of a space [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X=\{a,b\} }[/math] with the indiscrete topology, the set [math]\displaystyle{ A=\{a\} }[/math] is dense, but is not dense-in-itself.

The closure of any dense-in-itself set is a perfect set.

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

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.