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,
is dense in
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
containing
is
itself.
The of a topological space
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
Definition
A subset
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 ...
is said to be a of
if any of the following equivalent conditions are satisfied:
- 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 containing is itself.
- The closure of in is equal to That is,
- 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 is empty. That is,
- Every point in either belongs to 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
- For every 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 ...
of intersects that is,
- intersects every non-empty open subset of
and if
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
then this list can be extended to include:
- For every 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 ...
of intersects
- intersects every non-empty
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
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 of
in
is the
union of
and the set of all
limits of sequences of elements in
(its ''limit points''),
Then
is dense in
if
If
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,
then
is also dense in
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 and are dense open subset of a topological space If then the conclusion that the open set is dense in is immediate, so assume otherwise. Let is a non-empty open subset of so it remains to show that is also not empty. Because is dense in and is a non-empty open subset of their intersection is not empty. Similarly, because is a non-empty open subset of and is dense in their intersection is not empty. ]
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 ...