Karl Jones at 15:44, 14 October 2016

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Karl Jones: Created page with "In mathematics, a '''nowhere dense set''' in a topological space is a set whose closure has empty interior. == Descript..."

2016-10-14T15:25:43Z

Created page with "In mathematics, a '''nowhere dense set''' in a topological space is a set whose closure has empty interior. == Descript..."

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In [[mathematics]], a '''nowhere dense set''' in a [[topological space]] is a set whose [[Closure (topology)|closure]] has empty [[Interior (topology)|interior]].

== Description ==

In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. The order of operations is important. For example, the set of [[Rational number|rational numbers]], as a subset of R, has the property that the interior has an empty closure, but it is not nowhere dense; in fact it is [[Dense set|dense]] in R. Equivalently, a nowhere dense set is a set that is not dense in any nonempty open set.

The surrounding space matters: a set A may be nowhere dense when considered as a subspace of a topological space X but not when considered as a subspace of another topological space Y. A nowhere dense set is always dense in itself.

Every subset of a nowhere dense set is nowhere dense, and the [[Union (set theory)|union]] of [[Finite set|finitely]] many nowhere dense sets is nowhere dense. That is, the nowhere dense sets form an [[Ideal (set theory)|ideal of sets]], a suitable notion of [[negligible set]]. The union of [[Countable set|countably]] many nowhere dense sets, however, need not be nowhere dense. (Thus, the nowhere dense sets need not form a [[sigma-ideal]].) Instead, such a union is called a [[meagre set]] or a set of first category.

The concept is important to formulate the [[Baire category theorem]].

== See also ==

* [[Baire category theorem]]
* [[Baire space]]
* [[Closure (topology)]]
* [[Countable set]]
* [[Dense set]]
* [[Fat Cantor set]]
* [[Finite set]]
* [[Ideal (set theory)]]
* [[Interior (topology)]]
* [[Meagre set]]
* [[Negligible set]]
* [[Sigma-ideal]]
* [[Topological space]]

== External links ==

* [https://en.wikipedia.org/wiki/Nowhere_dense_set Nowhere dense set] @ Wikipedia

[[Category:Mathematics]]

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Nowhere dense set - Revision history

Karl Jones at 15:44, 14 October 2016

Karl Jones: /* External links */

Karl Jones: Created page with "In mathematics, a '''nowhere dense set''' in a topological space is a set whose closure has empty interior. == Descript..."