Dense-in-itself

In general topology, a subset $A$ of a topological space is said to be dense-in-itselfSteen & Seebach, p. 6Engelking, p. 25 or crowded{{cite journal |last1=Levy |first1=Ronnie |last2=Porter |first2=Jack |title=On Two questions of Arhangel'skii and Collins regarding submaximal spaces |journal=Topology Proceedings |date=1996 |volume=21 |pages=143–154 |url=http://topology.nipissingu.ca/tp/reprints/v21/tp21008.pdf}}{{cite web |url=https://www.researchgate.net/publication/228597275 |last1=Dontchev |first1=Julian |last2=Ganster |first2=Maximilian |last3=Rose |first3=David |date=1977 |title=α-Scattered spaces II}}

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 $\mathbb{R}$ . As an example that is dense-in-itself but not dense in its topological space, consider $\mathbb{Q} \cap [0,1]$ . This set is not dense in $\mathbb{R}$ 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.Engelking, 1.7.10, p. 59 In a dense-in-itself space, they include all open sets.Kuratowski, p. 78 In a dense-in-itself T₁ space they include all dense sets.Kuratowski, p. 78 However, spaces that are not T₁ may have dense subsets that are not dense-in-itself: for example in the dense-in-itself space $X=\{a,b\}$ with the indiscrete topology, the set $A=\{a\}$ 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.

Notes

References

{{cite book|last=Engelking|first=Ryszard| author-link=Ryszard Engelking|title=General Topology|publisher=Heldermann Verlag, Berlin|year=1989| isbn=3-88538-006-4}}
{{cite book|author =Kuratowski, K.|author-link =Kuratowski|publisher=Academic Press |year =1966|title=Topology Vol. I|isbn =012429202X}}
{{cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | year=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446}}

Category:Topology

Dense-in-itself

Examples

Properties

See also

Notes

References