In
mathematics, 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 po ...
is called nowhere dense or rare if its
closure has
empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the
topology on the space) anywhere. For example, the
integers are nowhere dense among the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
s, whereas an
open ball is not.
A countable union of nowhere dense sets is called a
meagre set. 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 the ...
, which is used in the proof of several fundamental result of
functional analysis.
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 po ...
is said to be ''dense'' in another set if the intersection is a dense subset 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 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
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of th ...
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 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
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''U ...
of nowhere dense sets is nowhere dense. Thus the nowhere dense sets form an
ideal of sets, 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 integ ...
. 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 calle ...
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
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 pl ...
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
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of th ...
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).
*