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In [[topology]] and related areas of [[mathematics]], a [[subset]] ''A'' of a [[topological space]] ''X'' is called '''dense''' (in ''X'') if every point ''x'' in ''X'' either belongs to ''A'' or is a [[limit point]] of ''A''.<ref name="CEIT">{{Citation |first=L. A. |last=Steen |first2=J. A. |last2=Seebach |title=[[Counterexamples in Topology]] |publisher=Dover |year=1995 |ISBN=0-486-68735-X}}</ref> Informally, for every point in ''X'', the point is either in ''A'' or arbitrarily "close" to a member of ''A'' - for instance, every [[real number]] is either a [[rational number]] or has one arbitrarily close to it (see [[Diophantine approximation]]).

Formally, a subset ''A'' of a topological space ''X'' is dense in ''X'' if for any point ''x'' in ''X'', any [[neighborhood (mathematics)|neighborhood]] of ''x'' contains at least one point from ''A''. Equivalently, ''A'' is dense in ''X'' if and only if the only [[closed set|closed subset]] of ''X'' containing ''A'' is ''X'' itself. This can also be expressed by saying that the [[closure (topology)|closure]] of ''A'' is ''X'', or that the [[interior (topology)|interior]] of the complement of ''A'' is empty.

The '''density''' of a topological space ''X'' is the least [[cardinality]] of a dense subset of ''X''.

==Density in metric spaces==
An alternative definition of dense set in the case of [[metric space]]s is the following. When the topology of ''X'' is given by a metric, the [[topological closure|closure]] <math>\displaystyle \overline{A}</math> of ''A'' in ''X'' is the union of ''A'' and the set of all limits of sequences of elements in ''A'' (its ''limit points''),
:<math>\overline{A} = A \cup \{ \lim_n a_n : \forall n \ge 0, \ a_n \in A \}</math>

Then ''A'' is dense in ''X'' if 
:<math>\overline{A} = X</math>

Note that <math>\scriptstyle  A \;\subseteq\; \{ \lim_n a_n:\, \forall n \;\ge\; 0,\, \ a_n \in A \}</math>. If <math>\displaystyle \{U_n\}</math> is a sequence of dense [[open set|open]] sets in a complete metric space, ''X'', then <math>\displaystyle \cap^{\infty}_{n=1} U_n</math> is also dense in ''X''.  This fact is one of the equivalent forms of the [[Baire category theorem]].

==Examples==
The [[real number]]s with the usual topology have the [[rational number]]s as a [[countable set|countable]] dense subset which shows that the [[cardinality]] of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The [[irrational number]]s are another dense subset which shows that a topological space may have several [[disjoint sets|disjoint]] dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality. Perhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any nonempty open set.

By the [[Weierstrass approximation theorem]], any given [[complex number|complex-valued]] [[continuous function]] defined on a [[closed interval]] [''a'',''b''] can be [[uniform convergence|uniformly approximated]] as closely as desired by a [[polynomial function]]. In other words, the polynomial functions are dense in the space C[''a'',''b''] of continuous complex-valued functions on the interval [''a'',''b''], equipped with the [[supremum norm]].

Every [[metric space]] is dense in its [[completion (metric space)|completion]].

== Properties ==
Every [[topological space]] is dense in itself. For a set ''X'' equipped with the [[discrete topology]] the whole space is the only dense set. Every non-empty subset of a set ''X'' equipped with the [[trivial topology]] is dense, and every topology for which every non-empty subset is dense must be trivial.

Denseness is [[transitive relation|transitive]]: Given three subsets ''A'', ''B'' and ''C'' of a topological space ''X'' with {{nowrap|''A'' ⊆ ''B'' ⊆ ''C''}} such that ''A'' is dense in ''B'' and ''B'' is dense in ''C'' (in the respective [[subspace topology]]) then ''A'' is also dense in ''C''.

The image of a dense subset under a [[surjective function|surjective]] continuous function is again dense. The density of a topological space (the least of the [[cardinality|cardinalities]] of its dense subsets) is a [[topological invariant]].

A topological space with a [[connected space|connected]] dense subset is necessarily connected itself.

[[Continuous function]]s into [[Hausdorff space]]s are determined by their values on dense subsets: if two continuous functions {{nowrap|''f'', ''g'' : ''X'' → ''Y''}} into a [[Hausdorff space]] ''Y'' agree on a dense subset of ''X'' then they agree on all of ''X''.

== Related notions ==
A point ''x'' of a subset ''A'' of a topological space ''X'' is called a [[limit point]] of ''A'' (in ''X'') if every neighbourhood of ''x'' also contains a point of ''A'' other than ''x'' itself, and an [[isolated point]] of ''A'' otherwise. A subset without isolated points is said to be [[dense-in-itself]].

A subset ''A'' of a topological space ''X'' is called [[nowhere dense set|nowhere dense]] (in ''X'') if there is no neighborhood in ''X'' on which ''A'' is dense. Equivalently, a subset of a topological space is nowhere dense if and only if the interior of its closure is empty. The interior of the complement of a nowhere dense set is always dense. The complement of a closed nowhere dense set is a dense open set. Given a topological space ''X'', a subset ''A'' of ''X'' that can be expressed as the union of [[countable set|countably many]] nowhere dense subsets of ''X'' is called [[meagre set|meagre]]. The [[rational numbers]], while dense in the [[real numbers]], are meagre as a subset of the reals.

A topological space with a [[countable]] dense subset is called [[separable space|separable]]. A topological space is a [[Baire space]] if and only if the intersection of countably many dense open sets is always dense. A topological space is called [[resolvable space|resolvable]] if it is the union of two disjoint dense subsets. More generally, a topological space is called κ-resolvable if it contains κ pairwise disjoint dense sets.

An [[embedding]] of a topological space ''X'' as a dense subset of a [[compact space]] is called a [[compactification (mathematics)|compactification]] of ''X''.

A [[linear operator]] between [[topological vector space]]s ''X'' and ''Y'' is said to be [[densely defined operator|densely defined]] if its [[domain (mathematics)|domain]] is a dense subset of ''X'' and if its [[range (mathematics)|range]] is contained within ''Y''. See also [[continuous linear extension]].

A topological space ''X'' is [[hyperconnected space|hyperconnected]] if and only if every nonempty open set is dense in ''X''. A topological space is [[submaximal space|submaximal]] if and only if every dense subset is open.

==See also==
*[[Dense order]]

==References==
===Notes===
{{Reflist}}

===General references===
{{refbegin}}
* {{cite book |author=Nicolas Bourbaki |authorlink=Nicolas Bourbaki |title=General Topology, Chapters 1&ndash;4 |series=Elements of Mathematics |year=1989 |origyear=1971 |publisher=[[Springer-Verlag]] |isbn=3-540-64241-2 |ref=bourbaki}}
* {{Citation | 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]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995}}
{{refend}}

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[[Category:General topology]]