TheInfoList

In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

and related areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

''A'' of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s are a dense subset of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s because every real number either is a rational number or has a rational number arbitrarily close to it (see
Diophantine approximation In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Fried ...
). Formally, $A$ is dense in $X$ if the smallest
closed subset In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
of $X$ containing $A$ is $X$ itself. The of a topological space $X$ is the least
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a dense subset of $X.$

# Definition

A subset $A$ of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
$X$ is said to be a of $X$ if any of the following equivalent conditions are satisfied:
1. The smallest
closed subset In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
of $X$ containing $A$ is $X$ itself.
2. The closure of $A$ in $X$ is equal to $X.$ That is, $\operatorname_X A = X.$
3. The
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
of the
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
of $A$ is empty. That is, $\operatorname_X \left(X \setminus A\right) = \varnothing.$
4. Every point in $X$ either belongs to $A$ or is a
limit point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
of $A.$
5. For every $x \in X,$ every
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
$U$ of $x$ intersects $A;$ that is, $U \cap A \neq \varnothing.$
6. $A$ intersects every non-empty open subset of $X.$
and if $\mathcal$ is a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of open sets for the topology on $X$ then this list can be extended to include:
1. For every $x \in X,$ every
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
$B \in \mathcal$ of $x$ intersects $A.$
2. $A$ intersects every non-empty $B \in \mathcal.$

## Density in metric spaces

An alternative definition of dense set in the case of
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s is the following. When the
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
of $X$ is given by a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
, the closure $\overline$ of $A$ in $X$ is the union of $A$ and the set of all limits of sequences of elements in $A$ (its ''limit points''), $\overline = A \cup \left\$ Then $A$ is dense in $X$ if $\overline = X.$ If $\left\$ is a sequence of dense
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
sets in a complete metric space, $X,$ then $\bigcap^_ U_n$ is also dense in $X.$ This fact is one of the equivalent forms of the
Baire category theorem The Baire category theorem (BCT) is an important result in general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set t ...
.

# Examples

The
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s with the usual topology have the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s as a
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
dense subset which shows that the
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s are another dense subset which shows that a topological space may have several 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.Suppose that $A$ and $B$ are dense open subset of a topological space $X.$ If $X = \varnothing$ then the conclusion that the open set $A \cap B$ is dense in $X$ is immediate, so assume otherwise. Let $U$ is a non-empty open subset of $X,$ so it remains to show that $U \cap \left(A \cap B\right)$ is also not empty. Because $A$ is dense in $X$ and $U$ is a non-empty open subset of $X,$ their intersection $U \cap A$ is not empty. Similarly, because $U \cap A$ is a non-empty open subset of $X$ and $B$ is dense in $X,$ their intersection $U \cap A \cap B$ is not empty. $\blacksquare$ The empty set is a dense subset of itself. But every dense subset of a non-empty space must also be non-empty. By the
Weierstrass approximation theorem Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
, any given
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
defined on a
closed interval In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, b The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math> can be uniformly approximated as closely as desired by a
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
. In other words, the polynomial functions are dense in the space $C$
, b The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math> of continuous complex-valued functions on the interval 
, b The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
equipped with the
supremum norm frame, The perimeter of the square is the set of points in R2 where the sup norm equals a fixed positive constant. In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded function Image:Bounded and ...
. Every
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is dense in its completion.

# Properties

Every
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
is a dense subset of itself. For a set $X$ equipped with the
discrete topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
, the whole space is the only dense subset. Every non-empty subset of a set $X$ equipped with the
trivial topologyIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
is dense, and every topology for which every non-empty subset is dense must be trivial. Denseness is
transitive Transitivity or transitive may refer to: Grammar * Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects * Transitive verb, a verb which takes an object * Transitive case, a grammatical case to mark arg ...
: Given three subsets $A, B$ and $C$ of a topological space $X$ with $A \subseteq B \subseteq C \subseteq X$ such that $A$ is dense in $B$ and $B$ is dense in $C$ (in the respective
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

) then $A$ is also dense in $C.$ The
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of a dense subset under a
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
function is again dense. The density of a topological space (the least of the
cardinalities In mathematics, the cardinality of a set (mathematics), set is a measure of the "number of Element (mathematics), elements" of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 1 ...
of its dense subsets) is a
topological invariantIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
. A topological space with a connected dense subset is necessarily connected itself. Continuous functions into
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

s are determined by their values on dense subsets: if two continuous functions $f, g : X \to Y$ into a
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

$Y$ agree on a dense subset of $X$ then they agree on all of $X.$ For metric spaces there are universal spaces, into which all spaces of given density can be embedded: a metric space of density $\alpha$ is isometric to a subspace of $C\left\left($
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
, \R\right), the space of real continuous functions on the product of $\alpha$ copies of the
unit interval In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
.

# Related notions

A point $x$ of a subset $A$ of a topological space $X$ is called a
limit point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
of $A$ (in $X$) if every neighbourhood of $x$ also contains a point of $A$ other than $x$ itself, and an
isolated point 400px, "0" is an isolated point of A = ∪ , 2In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
of $A$ otherwise. A subset without isolated points is said to be
dense-in-itselfIn general topology, a subset A of a topological space 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\subse ...
. A subset $A$ of a topological space $X$ is called 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 countably many nowhere dense subsets of $X$ is called 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. A topological space is a
Baire space In mathematics, a Baire space is a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and ...
if and only if the intersection of countably many dense open sets is always dense. A topological space is called resolvable if it is the union of two disjoint dense subsets. More generally, a topological space is called κ-resolvable for a
cardinal Cardinal or The Cardinal may refer to: Christianity * Cardinal (Catholic Church), a senior official of the Catholic Church * Cardinal (Church of England), two members of the College of Minor Canons of St. Paul's Cathedral Navigation * Cardina ...
κ if it contains κ pairwise disjoint dense sets. An
embedding In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of a topological space $X$ as a dense subset of a
compact space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is called a compactification of $X.$ A
linear operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
between
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s $X$ and $Y$ is said to be densely defined if its
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
is a dense subset of $X$ and if its
range Range may refer to: Geography * Range (geographic)A range, in geography, is a chain of hill A hill is a landform A landform is a natural or artificial feature of the solid surface of the Earth or other planetary body. Landforms together ...
is contained within $Y.$ See also
Continuous linear extensionIn functional analysis, it is often convenient to define a linear transformation on a complete space, complete, normed vector space X by first defining a linear transformation \mathsf on a dense set, dense subset of X and then extending \mathsf to th ...
. A topological space $X$ is hyperconnected if and only if every nonempty open set is dense in $X.$ A topological space is submaximal if and only if every dense subset is open. If $\left\left(X, d_X\right\right)$ is a metric space, then a non-empty subset $Y$ is said to be $\varepsilon$-dense if $\text x \in X, \text y \in Y \text d_X(x, y) \leq \varepsilon.$ One can then show that $D$ is dense in $\left\left(X, d_X\right\right)$ if and only if it is ε-dense for every $\varepsilon > 0.$