general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometr ...

, a subset

A

of a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

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 In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...

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 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 In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

). This set is dense-in-itself because every

neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

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

singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...

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 set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...

s. 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 space

X=\

with the indiscrete topology, the set

A=\

is dense, but is not dense-in-itself. The closure of any dense-it-itself set is a perfect set.Kuratowski, p. 77

Notes

References

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

Examples

Properties

See also

Notes

References