Nowhere dense set
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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. 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 reals, whereas the interval (0, 1) is not nowhere dense. 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 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 S 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 ...
X is said to be ''dense'' in another set U if the intersection S \cap U 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 U. S is or in X if S is not dense in any nonempty open subset U of X.
Expanding out the negation of density, it is equivalent that each nonempty open set U contains a nonempty open subset disjoint from S. It suffices to check either condition on a base for the topology on X. In particular, density nowhere in \R 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, \operatorname_X S, cannot contain any nonempty open set. This is the same as saying that the interior of the closure of S is empty; that is,
\operatorname_X \left(\operatorname_X S\right) = \varnothing.
Alternatively, the complement of the closure X \setminus \left(\operatorname_X S\right) must be a dense subset of X; in other words, the exterior of S is dense in X.


Properties

The notion of ''nowhere dense set'' is always relative to a given surrounding space. Suppose A\subseteq Y\subseteq X, where Y 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 X. The set A may be nowhere dense in X, but not nowhere dense in Y. Notably, a set is always dense in its own subspace topology. So if A is nonempty, it will not be nowhere dense as a subset of itself. However the following results hold: * If A is nowhere dense in Y, then A is nowhere dense in X. * If Y is open in X, then A is nowhere dense in Y if and only if A is nowhere dense in X. * If Y is dense in X, then A is nowhere dense in Y if and only if A is nowhere dense in X. 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 \Q is not nowhere dense in \R. 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 A\subseteq X 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 A).


Examples

* The set S=\ and its closure S\cup\ are nowhere dense in \R, 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 \R. * \R viewed as the horizontal axis in the Euclidean plane is nowhere dense in \R^2. * \Z is nowhere dense in \R but the rationals \Q are not (they are dense everywhere). * \Z \cup a, b) \cap \Q/math> is nowhere dense in \R: it is dense in the open interval (a,b), and in particular the interior of its closure is (a,b). * The empty set is nowhere dense. In a
discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
, the empty set is the nowhere dense set. * In a T1 space, any singleton set that is not an isolated point is nowhere dense. * A vector subspace of a
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 als ...
is either dense or nowhere dense.


Nowhere dense sets with positive measure

A nowhere dense set is not necessarily negligible in every sense. For example, if X is 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 ...
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
not only is it possible to have a dense set of
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
zero (such as the set of rationals), but it is also possible to have a nowhere dense set with positive measure. One such example is the
Smith–Volterra–Cantor set In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–V ...
. For another example (a variant of 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 ...
), remove from
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> all dyadic fractions, i.e. fractions of the form a/2^n in lowest terms for positive integers a, n \in \N, and the intervals around them: \left(a/2^n - 1/2^, a/2^n + 1/2^\right). Since for each n this removes intervals adding up to at most 1/2^, the nowhere dense set remaining after all such intervals have been removed has measure of at least 1/2 (in fact just over 0.535\ldots because of overlaps) and so in a sense represents the majority of the ambient space
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
This set is nowhere dense, as it is closed and has an empty interior: any interval (a, b) is not contained in the set since the dyadic fractions in (a, b) have been removed. Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than 1, although the measure cannot be exactly 1 (because otherwise the complement of its closure would be a nonempty open set with measure zero, which is impossible). For another simpler example, if U is any dense open subset of \R having finite
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
then \R \setminus U is necessarily a closed subset of \R having infinite Lebesgue measure that is also nowhere dense in \R (because its topological interior is empty). Such a dense open subset U of finite Lebesgue measure is commonly constructed when proving that the Lebesgue measure of the rational numbers \Q is 0. This may be done by choosing any
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
f : \N \to \Q (it actually suffices for f : \N \to \Q to merely be a
surjection In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
) and for every r > 0, letting U_r ~:=~ \bigcup_ \left(f(n) - r/2^n, f(n) + r/2^n\right) ~=~ \bigcup_ f(n) + \left(- r/2^n, r/2^n\right) (here, the
Minkowski sum In geometry, the Minkowski sum of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...
notation f(n) + \left(- r/2^n, r/2^n\right) := \left(f(n) - r/2^n, f(n) + r/2^n\right) was used to simplify the description of the intervals). The open subset U_r is dense in \R because this is true of its subset \Q and its Lebesgue measure is no greater than \sum_ 2 r / 2^n = 2 r. Taking the union of closed, rather than open, intervals produces the F-subset S_r ~:=~ \bigcup_ f(n) + \left r/2^n, r/2^n\right/math> that satisfies S_ \subseteq U_r \subseteq S_r \subseteq U_. Because \R \setminus S_r is a subset of the nowhere dense set \R \setminus U_r, it is also nowhere dense in \R. Because \R is a
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
, the set D := \bigcap_^ U_ = \bigcap_^ S_ is a dense subset of \R (which means that like its subset \Q, D cannot possibly be nowhere dense in \R) with 0 Lebesgue measure that is also a nonmeager subset of \R (that is, D is of the second category in \R), which makes \R \setminus D a comeager subset of \R whose interior in \R is also empty; however, \R \setminus D is nowhere dense in \R if and only if its in \R has empty interior. The subset \Q in this example can be replaced by any countable dense subset of \R and furthermore, even the set \R can be replaced by \R^n for any integer n > 0.


See also

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References


Bibliography

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External links


Some nowhere dense sets with positive measure
General topology de:Dichte Teilmenge#Nirgends dichte Teilmenge