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Dense set

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Subset whose closure is the whole space

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Intopology and related areas ofmathematics, asubsetA of atopological spaceX is said to bedense inX if every point ofX either belongs toA or else is arbitrarily "close" to a member ofA — for instance, therational numbers are a dense subset of thereal numbers because every real number either is a rational number or has a rational number arbitrarily close to it (seeDiophantine approximation). Formally, $A {\displaystyle A}$ is dense in $X {\displaystyle X}$ if the smallestclosed subset of $X {\displaystyle X}$ containing $A {\displaystyle A}$ is $X {\displaystyle X}$ itself.^[1]

Thedensity of a topological space $X {\displaystyle X}$ is the leastcardinality of a dense subset of $X . {\displaystyle X.}$

Definition

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A subset $A {\displaystyle A}$ of atopological space $X {\displaystyle X}$ is said to be adense subset of $X {\displaystyle X}$ if any of the following equivalent conditions are satisfied:

The smallestclosed subset of $X {\displaystyle X}$ containing $A {\displaystyle A}$ is $X {\displaystyle X}$ itself.
Theclosure of $A {\displaystyle A}$ in $X {\displaystyle X}$ is equal to $X . {\displaystyle X.}$ That is, $\operatorname {cl} _{X}A=X.$
Theinterior of thecomplement of $A {\displaystyle A}$ is empty. That is, $\operatorname {int} _{X}(X\setminus A)=\varnothing .$
Every point in $X {\displaystyle X}$ either belongs to $A {\displaystyle A}$ or is alimit point of $A . {\displaystyle A.}$
For every $x\in X,$ everyneighborhood $U {\displaystyle U}$ of $x {\displaystyle x}$ intersects $A; {\displaystyle A;}$ that is, $U\cap A\neq \varnothing .$
$A {\displaystyle A}$ intersects every non-empty open subset of $X . {\displaystyle X.}$

and if ${\mathcal {B}}$ is abasis of open sets for the topology on $X {\displaystyle X}$ then this list can be extended to include:

For every $x\in X,$ everybasicneighborhood $B\in {\mathcal {B}}$ of $x {\displaystyle x}$ intersects $A . {\displaystyle A.}$
$A {\displaystyle A}$ intersects every non-empty $B\in {\mathcal {B}}.$

Density in metric spaces

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An alternative definition of dense set in the case ofmetric spaces is the following. When thetopology of $X {\displaystyle X}$ is given by ametric, theclosure ${\overline {A}}$ of $A {\displaystyle A}$ in $X {\displaystyle X}$ is theunion of $A {\displaystyle A}$ and the set of alllimits of sequences of elements in $A {\displaystyle A}$ (itslimit points), ${\overline {A}}=A\cup \left\{\lim _{n\to \infty }a_{n}:a_{n}\in A{\text{ for all }}n\in \mathbb {N} \right\}$

Then $A {\displaystyle A}$ is dense in $X {\displaystyle X}$ if ${\overline {A}}=X.$

If $\left\{U_{n}\right\}$ is a sequence of denseopen sets in a complete metric space, $X, {\displaystyle X,}$ then ${\textstyle \bigcap _{n=1}^{\infty }U_{n}}$ is also dense in $X . {\displaystyle X.}$ This fact is one of the equivalent forms of theBaire category theorem.

Examples

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Thereal numbers with the usual topology have therational numbers as acountable dense subset which shows that thecardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. Theirrational numbers are another dense subset which shows that a topological space may have severaldisjoint 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 non-empty open set. The intersection of two dense open subsets of a topological space is again dense and open.^{[proof 1]} Theempty set is a dense subset of itself. But every dense subset of a non-empty space must also be non-empty.

By theWeierstrass approximation theorem, any givencomplex-valued continuous function defined on aclosed interval $[a,b]$ can beuniformly approximated as closely as desired by apolynomial 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 thesupremum norm.

Everymetric space is dense in itscompletion.

Properties

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Everytopological space is a dense subset of itself. For a set $X {\displaystyle X}$ equipped with thediscrete topology, the whole space is the only dense subset. Every non-empty subset of a set $X {\displaystyle X}$ equipped with thetrivial topology is dense, and every topology for which every non-empty subset is dense must be trivial.

Denseness istransitive: Given three subsets $A, B {\displaystyle A,B}$ and $C {\displaystyle C}$ of a topological space $X {\displaystyle X}$ with $A\subseteq B\subseteq C\subseteq X$ such that $A {\displaystyle A}$ is dense in $B {\displaystyle B}$ and $B {\displaystyle B}$ is dense in $C {\displaystyle C}$ (in the respectivesubspace topology) then $A {\displaystyle A}$ is also dense in $C . {\displaystyle C.}$

Theimage of a dense subset under asurjective continuous function is again dense. The density of a topological space (the least of thecardinalities of its dense subsets) is atopological invariant.

A topological space with aconnected dense subset is necessarily connected itself.

Continuous functions intoHausdorff spaces are determined by their values on dense subsets: if two continuous functions $f,g:X\to Y$ into aHausdorff space $Y {\displaystyle Y}$ agree on a dense subset of $X {\displaystyle X}$ then they agree on all of $X . {\displaystyle X.}$

For metric spaces there are universal spaces, into which all spaces of given density can beembedded: a metric space of density $\alpha$ is isometric to a subspace of $C\left([0,1]^{\alpha },\mathbb {R} \right),$ the space of real continuous functions on theproduct of $\alpha$ copies of theunit interval.^[2]

Related notions

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A point $x {\displaystyle x}$ of a subset $A {\displaystyle A}$ of a topological space $X {\displaystyle X}$ is called alimit point of $A {\displaystyle A}$ (in $X {\displaystyle X}$ ) if every neighbourhood of $x {\displaystyle x}$ also contains a point of $A {\displaystyle A}$ other than $x {\displaystyle x}$ itself, and anisolated point of $A {\displaystyle A}$ otherwise. A subset without isolated points is said to bedense-in-itself.

A subset $A {\displaystyle A}$ of a topological space $X {\displaystyle X}$ is callednowhere dense (in $X {\displaystyle X}$ ) if there is no neighborhood in $X {\displaystyle X}$ on which $A {\displaystyle A}$ is dense. Equivalently, a subset of a topological space is nowhere denseif 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 $X, {\displaystyle X,}$ a subset $A {\displaystyle A}$ of $X {\displaystyle X}$ that can be expressed as the union of countably many nowhere dense subsets of $X {\displaystyle X}$ is calledmeagre. 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 calledseparable. A topological space is aBaire space if and only if the intersection of countably many dense open sets is always dense. A topological space is calledresolvable if it is the union of two disjoint dense subsets. More generally, a topological space is called κ-resolvable for acardinal κ if it contains κ pairwise disjoint dense sets.

Anembedding of a topological space $X {\displaystyle X}$ as a dense subset of acompact space is called acompactification of $X . {\displaystyle X.}$

Alinear operator betweentopological vector spaces $X {\displaystyle X}$ and $Y {\displaystyle Y}$ is said to bedensely defined if itsdomain is a dense subset of $X {\displaystyle X}$ and if itsrange is contained within $Y . {\displaystyle Y.}$ See alsoContinuous linear extension.

A topological space $X {\displaystyle X}$ ishyperconnected if and only if every nonempty open set is dense in $X . {\displaystyle X.}$ A topological space issubmaximal if and only if every dense subset is open.

If $\left(X,d_{X}\right)$ is a metric space, then a non-empty subset $Y {\displaystyle Y}$ is said to be $\varepsilon$ -dense if $\forall x\in X,\;\exists y\in Y{\text{ such that }}d_{X}(x,y)\leq \varepsilon .$