TheInfoList

OR:

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 $S$ 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 ...
$X$ 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 to require 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 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, $\operatorname_X S,$ 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 $S$ is empty; that is,
$\operatorname_X \left\left(\operatorname_X S\right\right) = \varnothing.$
Alternatively, the complement of the closure $X \setminus \left\left(\operatorname_X S\right\right)$ 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'' 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 $X.$ 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 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 $A\subseteq X$ 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. * $\R$ 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). *
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 ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
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 al ...
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. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
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 ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
zero (such as the set of rationals), but it is also possible to have a nowhere dense set with positive measure. For one 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 introduced by German mathematician Georg Cantor in 1883. T ...
), remove from 
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> all dyadic fractions, i.e. fractions of the form $a/2^n$ in
lowest terms An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). I ...
for positive integers $a, n \in \N,$ and the intervals around them: $\left\left(a/2^n - 1/2^, a/2^n + 1/2^\right\right).$ 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. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
This set is nowhere dense, as it is closed and has an empty interior: any interval $\left(a, b\right)$ is not contained in the set since the dyadic fractions in $\left(a, b\right)$ 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 ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
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, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
$f : \N \to \Q$ (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 that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
) 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 (also known as dilation) 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'', i.e., the set : A + B = \. Analogously, the Minkowski ...
notation $f\left(n\right) + \left\left(- r/2^n, r/2^n\right\right) := \left\left(f\left(n\right) - r/2^n, f\left(n\right) + r/2^n\right\right)$ 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

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# Bibliography

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