TheInfoList

OR:

In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closin ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
''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 poi ...
''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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s are a dense subset of the
real number 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 ...
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, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated ...
). Formally, $A$ is dense in $X$ if the smallest
closed subset 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 c ...
of $X$ containing $A$ is $X$ itself. The of a topological space $X$ is the least
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of a dense subset of $X.$

# Definition

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 poi ...
$X$ is said to be a of $X$ if any of the following equivalent conditions are satisfied:
1. The smallest
closed subset 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 c ...
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 of the
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
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, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...
of $A.$
5. For every $x \in X,$ 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, ...
$U$ 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 basis, the net cost of an asset after adjusting for various tax-related items * Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
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, 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, ...
$B \in \mathcal$ 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, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s is the following. When the
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closin ...
of $X$ is given by a metric, 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 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.

# Examples

The
real number 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 ...
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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s as a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
dense subset which shows that the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s are another dense subset which shows that a topological space may have several 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 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, any given complex-valued
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
defined on a
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
supremum norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when ...
. Every
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
is dense in its completion.

# Properties

Every
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 poi ...
is a dense subset of itself. For a set $X$ equipped with the discrete topology, the whole space is the only dense subset. 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: 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 topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
) then $A$ is also dense in $C.$ The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensi ...
of a dense subset under a
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
continuous function is again dense. The density of a topological space (the least of the cardinalities of its dense subsets) is a topological invariant. A topological space with a connected dense subset is necessarily connected itself. Continuous functions into Hausdorff spaces are determined by their values on dense subsets: if two continuous functions $f, g : X \to Y$ into a Hausdorff space $Y$ 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 the space of real continuous functions on the product of $\alpha$ copies of the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...
.

# Related notions

A point $x$ of a subset $A$ of a topological space $X$ is called a
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...
of $A$ (in $X$) if every neighbourhood of $x$ also contains a point of $A$ other than $x$ itself, and an
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
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 (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 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 κ if it contains κ pairwise disjoint dense sets. An
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is g ...
of a topological space $X$ as a dense subset of a
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
is called a compactification of $X.$ A
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
between
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
s $X$ and $Y$ is said to be densely defined if its domain is a dense subset of $X$ and if its range is contained within $Y.$ See also Continuous linear extension. 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 $\forall x \in X, \; \exists y \in Y \text d_X(x, y) \leq \varepsilon.$ 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 $\varepsilon > 0.$