In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
of a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is called nowhere dense or rare if its
closure has
empty interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on the space) anywhere. For example, the
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s are nowhere dense among the
real
Real may refer to:
Currencies
* Argentine real
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Nature and science
* Reality, the state of things as they exist, rathe ...
s, whereas the
interval (0, 1) is not nowhere dense.
A countable union of nowhere dense sets is called a
meagre set
In the mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itse ...
. Meagre sets play an important role in the formulation of the
Baire category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that th ...
, which is used in the proof of several fundamental results of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
.
Definition
Density nowhere can be characterized in different (but equivalent) ways. The simplest definition is the one from density:
A subset of a topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is said to be ''dense'' in another set if the intersection is a dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''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 ra ...
of is or in if is not dense in any nonempty open subset of
Expanding out the negation of density, it is equivalent that each nonempty open set
contains a nonempty open subset disjoint from
It suffices to check either condition on a
base for the topology on
In particular, density nowhere in
is often described as being dense in no
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
.
Definition by closure
The second definition above is equivalent to requiring that the closure,
cannot contain any nonempty open set. This is the same as saying that the
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
of the
closure of
is empty; that is,
Alternatively, the complement of the closure
must be a dense subset of
in other words, the
exterior of
is dense in
Properties
The notion of ''nowhere dense set'' is always relative to a given surrounding space. Suppose
where
has the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
induced from
The set
may be nowhere dense in
but not nowhere dense in
Notably, a set is always dense in its own subspace topology. So if
is nonempty, it will not be nowhere dense as a subset of itself. However the following results hold:
* If
is nowhere dense in
then
is nowhere dense in
* If
is open in
, then
is nowhere dense in
if and only if
is nowhere dense in
* If
is dense in
, then
is nowhere dense in
if and only if
is nowhere dense in
A set is nowhere dense if and only if its closure is.
Every subset of a nowhere dense set is nowhere dense, and a finite
union of nowhere dense sets is nowhere dense. Thus the nowhere dense sets form an
ideal of sets
In the mathematical field of set theory, an ideal is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal ...
, a suitable notion of
negligible set
In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose.
As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral ...
. In general they do not form a
𝜎-ideal, as
meager set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called ...
s, which are the countable unions of nowhere dense sets, need not be nowhere dense. For example, the set
is not nowhere dense in
The
boundary of every open set and of every closed set is closed and nowhere dense. A closed set is nowhere dense if and only if it is equal to its boundary, if and only if it is equal to the boundary of some open set (for example the open set can be taken as the complement of the set). An arbitrary set
is nowhere dense if and only if it is a subset of the boundary of some open set (for example the open set can be taken as the
exterior of
).
Examples
* The set
and its closure
are nowhere dense in
since the closure has empty interior.
* The
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883.
Throu ...
is an uncountable nowhere dense set in
*
viewed as the horizontal axis in the Euclidean plane is nowhere dense in
*
is nowhere dense in
but the rationals
are not (they are dense everywhere).
*