In
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 ...
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 ...
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
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 ...
on the space) anywhere. For example, the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s are nowhere dense among the
reals, whereas an
open ball
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defi ...
is not.
A countable union of nowhere dense sets is called a
meagre 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 calle ...
. Meagre sets play an important role in the formulation of the
Baire category theorem, which is used in the proof of several fundamental result 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 (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
.
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 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 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 to require 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 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. Othe ...
.
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'' 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 ...
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, a suitable notion of
negligible set. In general they do not form a
𝜎-ideal, as
meager sets, 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
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
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.
*
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).
*