Nowhere dense set

In mathematics, a subset of a topological space 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 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, which is used in the proof of several fundamental results of functional analysis.

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 $X$ is said to be dense in another set $U$ if the intersection $S\cap U$ is a dense subset of $U.$ The set $S$ is nowhere dense or rare 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 $\mathbb {R}$ is often described as being dense in no open interval.

Definition by closure

The second definition above is equivalent to requiring that the closure, $\operatorname {cl} _{X}S,$ cannot contain any nonempty open set. This is the same as saying that the interior of the closure of $S$ is empty; that is,

$\operatorname {int} _{X}\left(\operatorname {cl} _{X}S\right)=\varnothing .$

Alternatively, the complement of the closure $X\setminus \left(\operatorname {cl} _{X}S\right)$ 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 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 $\mathbb {Q}$ is not nowhere dense in $\mathbb {R} .$

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 $S=\{1/n:n=1,2,...\}$ and its closure $S\cup \{0\}$ are nowhere dense in $\mathbb {R} ,$ since the closure has empty interior.
• The Cantor set is an uncountable nowhere dense set in $\mathbb {R} .$
• $\mathbb {R}$ viewed as the horizontal axis in the Euclidean plane is nowhere dense in $\mathbb {R} ^{2}.$
• $\mathbb {Z}$ is nowhere dense in $\mathbb {R}$ but the rationals $\mathbb {Q}$ are not (they are dense everywhere).
• $\mathbb {Z} \cup [(a,b)\cap \mathbb {Q} ]$ is not nowhere dense in $\mathbb {R}$: 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, the empty set is the only 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 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 $[0,1],$ not only is it possible to have a dense set of Lebesgue measure 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.

For another example (a variant of the Cantor set), remove from $[0,1]$ all dyadic fractions, i.e. fractions of the form $a/2^{n}$ in lowest terms for positive integers $a,n\in \mathbb {N} ,$ and the intervals around them: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(a/2^n - 1/2^{2n+1}, a/2^n + 1/2^{2n+1}\right).} Since for each $n$ this removes intervals adding up to at most Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2^{n+1},} the nowhere dense set remaining after all such intervals have been removed has measure of at least Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2} (in fact just over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.535\ldots} because of overlaps) and so in a sense represents the majority of the ambient space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, 1].} This set is nowhere dense, as it is closed and has an empty interior: any interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a, b)} is not contained in the set since the dyadic fractions in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a, b)} have been removed.

Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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).

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Such a dense open subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} of finite Lebesgue measure is commonly constructed when proving that the Lebesgue measure of the rational numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Q} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.} This may be done by choosing any bijection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \N \to \Q} (it actually suffices for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \N \to \Q} to merely be a surjection) and for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0,} letting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_r ~:=~ \bigcup_{n \in \N} \left(f(n) - r/2^n, f(n) + r/2^n\right) ~=~ \bigcup_{n \in \N} f(n) + \left(- r/2^n, r/2^n\right)} (here, the Minkowski sum notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_r} is dense in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} because this is true of its subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Q} and its Lebesgue measure is no greater than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n \in \N} 2 r / 2^n = 2 r.} Taking the union of closed, rather than open, intervals 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parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} (that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} is of the second category in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} ), which makes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R \setminus D} a comeager subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to 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interior. The subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Q} in this example can be replaced by any countable dense subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} and furthermore, even the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} can be replaced by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} for any integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n > 0.}

See also

• Baire space
• Smith–Volterra–Cantor set
• Meagre set

Bibliography

• Template:Bourbaki General Topology Part II Chapters 5-10
• Fremlin, D. H. (2002). Measure Theory. Lulu.com. ISBN 978-0-9566071-1-9.
• Haworth, R. C.; McCoy, R. A. (1977), Baire Spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Rudin Walter Functional Analysis
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels
• Template:Willard General Topology

External links

• Some nowhere dense sets with positive measure