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Nowhere dense set

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a <a href="Set_(mathematics)" title="Set (mathematics)">subset</a> of a <a href="Topological_space" title="Topological space">topological space</a> is called nowhere dense<a href="#cite_note-FOOTNOTEBourbaki1989ch._IX,_section_5.1-1">[1]</a><a href="#cite_note-FOOTNOTEWillard2004Problem_4G-2">[2]</a> or rare<a href="#cite_note-FOOTNOTENariciBeckenstein2011section_11.5,_pp._387-389-3">[3]</a> if its <a href="Closure_(topology)" title="Closure (topology)">closure</a> has <a href="Empty_set" title="Empty set">empty</a> <a href="Interior_(topology)" title="Interior (topology)">interior</a>. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the <a href="Topological_space#Definitions" title="Topological space">topology</a> on the space) anywhere. For example, the <a href="Integer" title="Integer">integers</a> are nowhere dense among the <a href="Real_number" title="Real number">reals</a>, whereas the <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> (0, 1) is not nowhere dense.

A countable union of nowhere dense sets is called a <a href="Meagre_set" title="Meagre set">meagre set</a>. Meagre sets play an important role in the formulation of the <a href="Baire_category_theorem" title="Baire category theorem">Baire category theorem</a>, which is used in the proof of several fundamental results of <a href="Functional_analysis" title="Functional analysis">functional analysis</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Density nowhere can be characterized in different (but equivalent) ways. The simplest definition is the one from density:

<blockquote>A subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> of a <a href="Topological_space" title="Topological space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is said to be dense in another set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> if the intersection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\cap U}">

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<annotation encoding="application/x-tex">{\displaystyle S\cap U}</annotation>

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</math><img src="./418b657b6b9cd14f39ff73cea5c61f619c9a039e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.864ex; height:2.176ex;" alt="{\displaystyle S\cap U}" loading="lazy"> is a <a href="Dense_set" title="Dense set">dense subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}">

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<annotation encoding="application/x-tex">{\displaystyle U.}</annotation>

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</math><img src="./7a305ef479ab152035f334467a2c314baa23eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="{\displaystyle U.}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is nowhere dense or rare in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is not dense in any nonempty open subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

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<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

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</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> </blockquote>

Expanding out the negation of density, it is equivalent that each nonempty open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> contains a nonempty open subset disjoint from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S.}">

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<annotation encoding="application/x-tex">{\displaystyle S.}</annotation>

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</math><img src="./23bbb1f0f6ebdfa78b4fed06049640f7386bb44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle S.}" loading="lazy"><a href="#cite_note-FOOTNOTEFremlin20023A3F(a)-4">[4]</a> It suffices to check either condition on a <a href="Base_(topology)" title="Base (topology)">base</a> for the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

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<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

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</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> In particular, density nowhere in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> is often described as being dense in no <a href="Open_Interval" class="mw-redirect" title="Open Interval">open interval</a>.<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Definition_by_closure">Definition by closure</h3></div>

The second definition above is equivalent to requiring that the closure, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}S,}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}S,}</annotation>

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</math><img src="./714529f59580ac9b7a3b5d0f4688370907ac09fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.845ex; height:2.509ex;" alt="{\displaystyle \operatorname {cl} _{X}S,}" loading="lazy"> cannot contain any nonempty open set.<a href="#cite_note-7">[7]</a> This is the same as saying that the <a href="Interior_(topology)" title="Interior (topology)">interior</a> of the <a href="Closure_(topology)" title="Closure (topology)">closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is empty; that is,<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {int} _{X}\left(\operatorname {cl} _{X}S\right)=\varnothing .}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {int} _{X}\left(\operatorname {cl} _{X}S\right)=\varnothing .}</annotation>

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</math><img src="./aebaab93a227b4e5980302ffeaa05011b352e6df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.037ex; height:2.843ex;" alt="{\displaystyle \operatorname {int} _{X}\left(\operatorname {cl} _{X}S\right)=\varnothing .}" loading="lazy"><a href="#cite_note-:0-8">[8]</a><a href="#cite_note-FOOTNOTERudin199141-9">[9]</a> </blockquote>Alternatively, the complement of the closure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\setminus \left(\operatorname {cl} _{X}S\right)}">

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<annotation encoding="application/x-tex">{\displaystyle X\setminus \left(\operatorname {cl} _{X}S\right)}</annotation>

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</math><img src="./547e6065898aea68e92d1eeaab07232a40a7fe7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.182ex; height:2.843ex;" alt="{\displaystyle X\setminus \left(\operatorname {cl} _{X}S\right)}" loading="lazy"> must be a dense subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X;}">

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<annotation encoding="application/x-tex">{\displaystyle X;}</annotation>

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</math><img src="./382fc9d9db960b232f5960d73b4a4762c7a047e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X;}" loading="lazy"><a href="#cite_note-FOOTNOTEFremlin20023A3F(a)-4">[4]</a><a href="#cite_note-:0-8">[8]</a> in other words, the <a href="Exterior_(topology)" class="mw-redirect" title="Exterior (topology)">exterior</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

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<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

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</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

The notion of nowhere dense set is always relative to a given surrounding space. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq Y\subseteq X,}">

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<annotation encoding="application/x-tex">{\displaystyle A\subseteq Y\subseteq X,}</annotation>

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</math><img src="./7923633c21900a2ea31cdd2502f60983a7ebacec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.34ex; height:2.509ex;" alt="{\displaystyle A\subseteq Y\subseteq X,}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> has the <a href="Subspace_topology" title="Subspace topology">subspace topology</a> induced from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

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<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

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</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> may be nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

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<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

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</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> but not nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">

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<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>

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</math><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"> Notably, a set is always dense in its own subspace topology. So if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is nonempty, it will not be nowhere dense as a subset of itself. However the following results hold:<a href="#cite_note-FOOTNOTENariciBeckenstein2011Theorem_11.5.4-10">[10]</a><a href="#cite_note-FOOTNOTEHaworthMcCoy1977Proposition_1.3-11">[11]</a>

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y,}">

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<annotation encoding="application/x-tex">{\displaystyle Y,}</annotation>

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</math><img src="./a3765557b7effa1a5f2f4dce9c80a25973b7009f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

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<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

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</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is open in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

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<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

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</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"></li></ul>

A set is nowhere dense if and only if its closure is.<a href="#cite_note-FOOTNOTEBourbaki1989ch._IX,_section_5.1-1">[1]</a>

Every subset of a nowhere dense set is nowhere dense, and a finite <a href="Union_(set_theory)" title="Union (set theory)">union</a> of nowhere dense sets is nowhere dense.<a href="#cite_note-FOOTNOTEFremlin20023A3F(c)-12">[12]</a><a href="#cite_note-FOOTNOTEWillard2004Problem_25A-13">[13]</a> Thus the nowhere dense sets form an <a href="Ideal_of_sets" class="mw-redirect" title="Ideal of sets">ideal of sets</a>, a suitable notion of <a href="Negligible_set" title="Negligible set">negligible set</a>. In general they do not form a <a href="Sigma-ideal" title="Sigma-ideal">𝜎-ideal</a>, as <a href="Meager_set" class="mw-redirect" title="Meager set">meager sets</a>, which are the countable unions of nowhere dense sets, need not be nowhere dense. For example, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> is not nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation>

</semantics>

</math><img src="./dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}" loading="lazy">

The <a href="Boundary_(topology)" title="Boundary (topology)">boundary</a> of every open set and of every closed set is closed and nowhere dense.<a href="#cite_note-FOOTNOTENariciBeckenstein2011Example_11.5.3(e)-14">[14]</a><a href="#cite_note-FOOTNOTEWillard2004Problem_4G-2">[2]</a> A closed set is nowhere dense if and only if it is equal to its boundary,<a href="#cite_note-FOOTNOTENariciBeckenstein2011Example_11.5.3(e)-14">[14]</a> if and only if it is equal to the boundary of some open set<a href="#cite_note-FOOTNOTEWillard2004Problem_4G-2">[2]</a> (for example the open set can be taken as the complement of the set). An arbitrary set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq X}</annotation>

</semantics>

</math><img src="./1dce86da0107830a9a97287f9486d9b4ff022875.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.822ex; height:2.343ex;" alt="{\displaystyle A\subseteq X}" loading="lazy"> 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 <a href="Exterior_(topology)" class="mw-redirect" title="Exterior (topology)">exterior</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">).

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=\{1/n:n=1,2,...\}}</annotation>

</semantics>

</math><img src="./85f27682930d964d163ab75e66eb4a45b513abb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.567ex; height:2.843ex;" alt="{\displaystyle S=\{1/n:n=1,2,...\}}" loading="lazy"> and its closure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\cup \{0\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\cup \{0\}}</annotation>

</semantics>

</math><img src="./8406c62db0f47aaa1094feced744abe6ab1f82ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.569ex; height:2.843ex;" alt="{\displaystyle S\cup \{0\}}" loading="lazy"> are nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation>

</semantics>

</math><img src="./0522388d36b55de7babe4bbfc49475eaf590c2bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}" loading="lazy"> since the closure has empty interior.</li>

<li>The <a href="Cantor_set" title="Cantor set">Cantor set</a> is an uncountable nowhere dense set in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation>

</semantics>

</math><img src="./dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}" loading="lazy"></li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> viewed as the horizontal axis in the Euclidean plane is nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}.}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}.}</annotation>

</semantics>

</math><img src="./066b155c535a38739cc0c4b288324cbb7a4a227a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}.}" loading="lazy"></li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> is nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> but the rationals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> are not (they are dense everywhere).</li>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \cup [(a,b)\cap \mathbb {Q} ]}</annotation>

</semantics>

</math><img src="./08a985c33820fb31020e00a7e6016492e56e2e2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.888ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} \cup [(a,b)\cap \mathbb {Q} ]}" loading="lazy"> is not nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">: it is dense in the open interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b),}</annotation>

</semantics>

</math><img src="./0066e8fc90702a659ee69bff970050aba59ee02c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.717ex; height:2.843ex;" alt="{\displaystyle (a,b),}" loading="lazy"> and in particular the interior of its closure is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b).}</annotation>

</semantics>

</math><img src="./b828bde62741f9a659d25b1264068c81c3d0b946.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.717ex; height:2.843ex;" alt="{\displaystyle (a,b).}" loading="lazy"></li>

<li>The empty set is nowhere dense. In a <a href="Discrete_space" title="Discrete space">discrete space</a>, the empty set is the only nowhere dense set.<a href="#cite_note-FOOTNOTENariciBeckenstein2011Example_11.5.3(a)-15">[15]</a></li>

<li>In a <a href="T1_space" title="T1 space">T1 space</a>, any singleton set that is not an <a href="Isolated_point" title="Isolated point">isolated point</a> is nowhere dense.</li>

<li>A <a href="Vector_subspace" class="mw-redirect" title="Vector subspace">vector subspace</a> of a <a href="Topological_vector_space" title="Topological vector space">topological vector space</a> is either dense or nowhere dense.<a href="#cite_note-FOOTNOTENariciBeckenstein2011Example_11.5.3(f)-16">[16]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Nowhere_dense_sets_with_positive_measure">Nowhere dense sets with positive measure</h2></div>

A nowhere dense set is not necessarily negligible in every sense. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the <a href="Unit_interval" title="Unit interval">unit interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1],}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1],}</annotation>

</semantics>

</math><img src="./971caee396752d8bf56711f55d2c3b1207d4a236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.299ex; height:2.843ex;" alt="{\displaystyle [0,1],}" loading="lazy"> not only is it possible to have a dense set of <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> 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 <a href="Smith%E2%80%93Volterra%E2%80%93Cantor_set" title="Smith–Volterra–Cantor set">Smith–Volterra–Cantor set</a>.

For another example (a variant of the <a href="Cantor_set" title="Cantor set">Cantor set</a>), remove from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

</math><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy"> all <a href="Dyadic_fraction" class="mw-redirect" title="Dyadic fraction">dyadic fractions</a>, i.e. fractions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a/2^{n}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a/2^{n}}</annotation>

</semantics>

</math><img src="./3f0061edb4167a9459e3321e35eeb68304626e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle a/2^{n}}" loading="lazy"> in <a href="Lowest_terms" class="mw-redirect" title="Lowest terms">lowest terms</a> for positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,n\in \mathbb {N} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,n\in \mathbb {N} ,}</annotation>

</semantics>

</math><img src="./94bf92c9d2d4c5ca34e33cb122ad530809e9e74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.824ex; height:2.509ex;" alt="{\displaystyle a,n\in \mathbb {N} ,}" loading="lazy"> and the intervals around them: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a/2^{n}-1/2^{2n+1},a/2^{n}+1/2^{2n+1}\right).}">

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(a/2^{n}-1/2^{2n+1},a/2^{n}+1/2^{2n+1}\right).}</annotation>

</semantics>

</math><img src="./372cd265e1ee09600e098a40e21510fc3d819b6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.682ex; height:3.343ex;" alt="{\displaystyle \left(a/2^{n}-1/2^{2n+1},a/2^{n}+1/2^{2n+1}\right).}" loading="lazy">

Since for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> this removes intervals adding up to at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2^{n+1},}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/2^{n+1},}</annotation>

</semantics>

</math><img src="./87140c977a481f1ee8aa99495b2d898b6e062182.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.453ex; height:3.176ex;" alt="{\displaystyle 1/2^{n+1},}" loading="lazy"> the nowhere dense set remaining after all such intervals have been removed has measure of at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/2}</annotation>

</semantics>

</math><img src="./e308a3a46b7fdce07cc09dcab9e8d8f73e37d935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 1/2}" loading="lazy"> (in fact just over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.535\ldots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0.535\ldots }</annotation>

</semantics>

</math><img src="./1a53f8f6c2d93e75d79c250fc2980bd3fc53d152.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.407ex; height:2.176ex;" alt="{\displaystyle 0.535\ldots }" loading="lazy"> because of overlaps<a href="#cite_note-17">[17]</a>) and so in a sense represents the majority of the ambient space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1].}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1].}</annotation>

</semantics>

</math><img src="./8786b5ef9daedb24adb59e7825c4096d99a99648.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.299ex; height:2.843ex;" alt="{\displaystyle [0,1].}" loading="lazy">

This set is nowhere dense, as it is closed and has an empty interior: any interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation>

</semantics>

</math><img src="./d7e5710198f33b00695903460983021e75860e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}" loading="lazy"> is not contained in the set since the dyadic fractions in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation>

</semantics>

</math><img src="./d7e5710198f33b00695903460983021e75860e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}" loading="lazy"> have been removed.

Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1,}</annotation>

</semantics>

</math><img src="./9cc5fd8163a83100c5330622e9e317fa4e872403.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 1,}" loading="lazy"> 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).<a href="#cite_note-18">[18]</a>

For another simpler example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> is any dense open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> having finite <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus U}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus U}</annotation>

</semantics>

</math><img src="./385fdc80a1d1f01edf6aad3bb96dc030031c099c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.655ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \setminus U}" loading="lazy"> is necessarily a closed subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> having infinite Lebesgue measure that is also nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> (because its topological interior is empty). Such a dense open subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> of finite Lebesgue measure is commonly constructed when proving that the Lebesgue measure of the rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0.}</annotation>

</semantics>

</math><img src="./916e773e0593223c306a3e6852348177d1934962.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 0.}" loading="lazy"> This may be done by choosing any <a href="Bijection" title="Bijection">bijection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {N} \to \mathbb {Q} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {N} \to \mathbb {Q} }</annotation>

</semantics>

</math><img src="./02fc4b63470cec8a6501c54e2769f82ce7b6c03d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.316ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {N} \to \mathbb {Q} }" loading="lazy"> (it actually suffices for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {N} \to \mathbb {Q} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {N} \to \mathbb {Q} }</annotation>

</semantics>

</math><img src="./02fc4b63470cec8a6501c54e2769f82ce7b6c03d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.316ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {N} \to \mathbb {Q} }" loading="lazy"> to merely be a <a href="Surjection" class="mw-redirect" title="Surjection">surjection</a>) and for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r>0,}</annotation>

</semantics>

</math><img src="./72ee1bdad381ab757caae14bc628dd1f9005fa5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.956ex; height:2.509ex;" alt="{\displaystyle r>0,}" loading="lazy"> letting

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{r}~:=~\bigcup _{n\in \mathbb {N} }\left(f(n)-r/2^{n},f(n)+r/2^{n}\right)~=~\bigcup _{n\in \mathbb {N} }f(n)+\left(-r/2^{n},r/2^{n}\right)}</annotation>

</semantics>

</math>

(here, the <a href="Minkowski_sum" class="mw-redirect" title="Minkowski sum">Minkowski sum</a> notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)+\left(-r/2^{n},r/2^{n}\right):=\left(f(n)-r/2^{n},f(n)+r/2^{n}\right)}">

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(n)+\left(-r/2^{n},r/2^{n}\right):=\left(f(n)-r/2^{n},f(n)+r/2^{n}\right)}</annotation>

</semantics>

</math><img src="./f7e9aa0d0750f621d805e7de6a5429c429907c6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.577ex; height:2.843ex;" alt="{\displaystyle f(n)+\left(-r/2^{n},r/2^{n}\right):=\left(f(n)-r/2^{n},f(n)+r/2^{n}\right)}" loading="lazy"> was used to simplify the description of the intervals).

The open subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{r}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{r}}</annotation>

</semantics>

</math><img src="./3eaec4ad37894dbbbf185dad9af9dcb365ab8916.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.561ex; height:2.509ex;" alt="{\displaystyle U_{r}}" loading="lazy"> is dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> because this is true of its subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> and its Lebesgue measure is no greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\in \mathbb {N} }2r/2^{n}=2r.}">

</mrow>

</munder>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n\in \mathbb {N} }2r/2^{n}=2r.}</annotation>

</semantics>

</math><img src="./f5078d41146bddf7cfb8f4670a158168f13e4987.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.453ex; height:5.676ex;" alt="{\displaystyle \sum _{n\in \mathbb {N} }2r/2^{n}=2r.}" loading="lazy">

Taking the union of closed, rather than open, intervals produces the <a href="F%CF%83_set" title="Fσ set">F𝜎-subset</a>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{r}~:=~\bigcup _{n\in \mathbb {N} }f(n)+\left[-r/2^{n},r/2^{n}\right]}</annotation>

</semantics>

</math>

that satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{r/2}\subseteq U_{r}\subseteq S_{r}\subseteq U_{2r}.}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{r/2}\subseteq U_{r}\subseteq S_{r}\subseteq U_{2r}.}</annotation>

</semantics>

</math><img src="./c87f9002c794617d9085a769a520ef13fa52d418.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.328ex; height:3.009ex;" alt="{\displaystyle S_{r/2}\subseteq U_{r}\subseteq S_{r}\subseteq U_{2r}.}" loading="lazy"> Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus S_{r}}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus S_{r}}</annotation>

</semantics>

</math><img src="./5a1ce4887fa5afae3421e1cc428045bed1b3af2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.271ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \setminus S_{r}}" loading="lazy"> is a subset of the nowhere dense set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus U_{r},}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus U_{r},}</annotation>

</semantics>

</math><img src="./957251694f8167a8b463017edfe2c29feab995ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.081ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \setminus U_{r},}" loading="lazy"> it is also nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation>

</semantics>

</math><img src="./dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}" loading="lazy">

Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> is a <a href="Baire_space" title="Baire space">Baire space</a>, the set

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D:=\bigcap _{m=1}^{\infty }U_{1/m}=\bigcap _{m=1}^{\infty }S_{1/m}}</annotation>

</semantics>

</math>

is a dense subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> (which means that like its subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ,}</annotation>

</semantics>

</math><img src="./91185244fbdded6ea99a5e9e6603299128b10928.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} ,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> cannot possibly be nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">) with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> Lebesgue measure that is also a <a href="Nonmeager_set" class="mw-redirect" title="Nonmeager set">nonmeager subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is of the <a href="Second_category" class="mw-redirect" title="Second category">second category</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">), which makes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus D}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus D}</annotation>

</semantics>

</math><img src="./d6abbe8f86dd649321ef42acbf7a4af400e4e889.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.797ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \setminus D}" loading="lazy"> a <a href="Comeager_set" class="mw-redirect" title="Comeager set">comeager subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> whose interior in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> is also empty; however, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus D}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus D}</annotation>

</semantics>

</math><img src="./d6abbe8f86dd649321ef42acbf7a4af400e4e889.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.797ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \setminus D}" loading="lazy"> is nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> if and only if its closure in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> has empty interior.

The subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> in this example can be replaced by any countable dense subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> and furthermore, even the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> can be replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> for any integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>0.}</annotation>

</semantics>

</math><img src="./57ab13b00b6f8bb19aea0dcfcc78f18a2b6cbffc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.302ex; height:2.176ex;" alt="{\displaystyle n>0.}" loading="lazy">

<ul><li><a href="Baire_space" title="Baire space">Baire space</a> – Concept in topology</li>

<li><a href="Smith%E2%80%93Volterra%E2%80%93Cantor_set" title="Smith–Volterra–Cantor set">Smith–Volterra–Cantor set</a> – Set of real numbers in mathematics</li>

<li><a href="Meagre_set" title="Meagre set">Meagre set</a> – "Small" subset of a topological space</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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</style><div class="reflist">

<li id="cite_note-FOOTNOTEBourbaki1989ch._IX,_section_5.1-1">^ <a href="#cite_ref-FOOTNOTEBourbaki1989ch._IX,_section_5.1_1-0">a</a> <a href="#cite_ref-FOOTNOTEBourbaki1989ch._IX,_section_5.1_1-1">b</a> <a href="#CITEREFBourbaki1989">Bourbaki 1989</a>, ch. IX, section 5.1.

</li>

<li id="cite_note-FOOTNOTEWillard2004Problem_4G-2">^ <a href="#cite_ref-FOOTNOTEWillard2004Problem_4G_2-0">a</a> <a href="#cite_ref-FOOTNOTEWillard2004Problem_4G_2-1">b</a> <a href="#cite_ref-FOOTNOTEWillard2004Problem_4G_2-2">c</a> <a href="#CITEREFWillard2004">Willard 2004</a>, Problem 4G.

</li>

<li id="cite_note-FOOTNOTENariciBeckenstein2011section_11.5,_pp._387-389-3"><a href="#cite_ref-FOOTNOTENariciBeckenstein2011section_11.5,_pp._387-389_3-0">^</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, section 11.5, pp. 387-389.

</li>

<li id="cite_note-FOOTNOTEFremlin20023A3F(a)-4">^ <a href="#cite_ref-FOOTNOTEFremlin20023A3F(a)_4-0">a</a> <a href="#cite_ref-FOOTNOTEFremlin20023A3F(a)_4-1">b</a> <a href="#CITEREFFremlin2002">Fremlin 2002</a>, 3A3F(a).

</li>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFOxtoby1980" class="citation book cs1">Oxtoby, John C. (1980). Measure and Category (2nd ed.). New York: Springer-Verlag. pp. 1–2. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-90508-1</bdi>. <q>A set is nowhere dense if it is dense in no interval</q></cite>; although note that Oxtoby later gives the interior-of-closure definition on page 40.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFNatanson1955" class="citation book cs1 cs1-prop-long-vol">Natanson, Israel P. (1955). <a rel="nofollow" class="external text" href="http://hdl.handle.net/2027/mdp.49015000681685">Teoria functsiy veshchestvennoy peremennoy</a> [Theory of functions of a real variable]. Vol. I (Chapters 1-9). Translated by Boron, Leo F. New York: Frederick Ungar. p. 88. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fmdp.49015000681685">2027/mdp.49015000681685</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/54-7420">54-7420</a>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFSteenSeebach_Jr.1995" class="citation book cs1">Steen, Lynn Arthur; Seebach Jr., J. Arthur (1995). Counterexamples in Topology (Dover republication of Springer-Verlag 1978 ed.). New York: Dover. p. 7. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-68735-3</bdi>. <q>A subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is said to be nowhere dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> if no nonempty open set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A}}.}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {A}}.}</annotation>

</semantics>

</math><img src="./a0ea102747993704ea8507ea27d4691a485727b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.505ex; height:3.009ex;" alt="{\displaystyle {\overline {A}}.}" loading="lazy"></q></cite>

</li>

<li id="cite_note-:0-8">^ <a href="#cite_ref-:0_8-0">a</a> <a href="#cite_ref-:0_8-1">b</a> <cite id="CITEREFGamelin1999" class="citation book cs1">Gamelin, Theodore W. (1999). Introduction to Topology (2nd ed.). Mineola: Dover. pp. 36–37. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-40680-6</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTERudin199141-9"><a href="#cite_ref-FOOTNOTERudin199141_9-0">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 41.

</li>

<li id="cite_note-FOOTNOTENariciBeckenstein2011Theorem_11.5.4-10"><a href="#cite_ref-FOOTNOTENariciBeckenstein2011Theorem_11.5.4_10-0">^</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, Theorem 11.5.4.

</li>

<li id="cite_note-FOOTNOTEHaworthMcCoy1977Proposition_1.3-11"><a href="#cite_ref-FOOTNOTEHaworthMcCoy1977Proposition_1.3_11-0">^</a> <a href="#CITEREFHaworthMcCoy1977">Haworth & McCoy 1977</a>, Proposition 1.3.

</li>

<li id="cite_note-FOOTNOTEFremlin20023A3F(c)-12"><a href="#cite_ref-FOOTNOTEFremlin20023A3F(c)_12-0">^</a> <a href="#CITEREFFremlin2002">Fremlin 2002</a>, 3A3F(c).

</li>

<li id="cite_note-FOOTNOTEWillard2004Problem_25A-13"><a href="#cite_ref-FOOTNOTEWillard2004Problem_25A_13-0">^</a> <a href="#CITEREFWillard2004">Willard 2004</a>, Problem 25A.

</li>

<li id="cite_note-FOOTNOTENariciBeckenstein2011Example_11.5.3(e)-14">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein2011Example_11.5.3(e)_14-0">a</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011Example_11.5.3(e)_14-1">b</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, Example 11.5.3(e).

</li>

<li id="cite_note-FOOTNOTENariciBeckenstein2011Example_11.5.3(a)-15"><a href="#cite_ref-FOOTNOTENariciBeckenstein2011Example_11.5.3(a)_15-0">^</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, Example 11.5.3(a).

</li>

<li id="cite_note-FOOTNOTENariciBeckenstein2011Example_11.5.3(f)-16"><a href="#cite_ref-FOOTNOTENariciBeckenstein2011Example_11.5.3(f)_16-0">^</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, Example 11.5.3(f).

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.se16.info/hgb/nowhere.htm">"Some nowhere dense sets with positive measure and a strictly monotonic continuous function with a dense set of points with zero derivative"</a>.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFFolland1984" class="citation book cs1">Folland, G. B. (1984). <a rel="nofollow" class="external text" href="http://hdl.handle.net/2027/mdp.49015000929258">Real analysis: modern techniques and their applications</a>. New York: John Wiley & Sons. p. 41. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fmdp.49015000929258">2027/mdp.49015000929258</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-80958-6</bdi>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFBourbaki1989" class="citation book cs1"><a href="Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1989) [1967]. General Topology 2: Chapters 5–10 [Topologie Générale]. <a href="%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Éléments de mathématique</a>. Vol. 4. Berlin New York: Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-64563-4</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/246032063">246032063</a>.</cite></li>

<li><cite id="CITEREFFremlin2002" class="citation book cs1">Fremlin, D. H. (2002). Measure Theory. Lulu.com. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-9566071-1-9</bdi>.</cite></li>

<li><cite id="CITEREFHaworthMcCoy1977" class="citation cs2">Haworth, R. C.; McCoy, R. A. (1977), <a rel="nofollow" class="external text" href="http://eudml.org/doc/268479">Baire Spaces</a>, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk</cite></li>

<li><cite id="CITEREFKhaleelulla1982" class="citation book cs1">Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. <a href="Lecture_Notes_in_Mathematics" title="Lecture Notes in Mathematics">Lecture Notes in Mathematics</a>. Vol. 936. Berlin, Heidelberg, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-11565-6</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/8588370">8588370</a>.</cite></li>

<li><cite id="CITEREFNariciBeckenstein2011" class="citation book cs1">Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1584888666</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/144216834">144216834</a>.</cite></li>

<li><cite id="CITEREFRudin1991" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-054236-5</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21163277">21163277</a>.</cite></li>

<li><cite id="CITEREFSchaeferWolff1999" class="citation book cs1"><a href="Helmut_H._Schaefer" title="Helmut H. Schaefer">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4612-7155-0</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840278135">840278135</a>.</cite></li>

<li><cite id="CITEREFTrèves2006" class="citation book cs1"><a href="Fran%C3%A7ois_Tr%C3%A8ves" title="François Trèves">Trèves, François</a> (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-45352-1</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/853623322">853623322</a>.</cite></li>

<li><cite id="CITEREFWillard2004" class="citation book cs1">Willard, Stephen (2004) [1970]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-o8xJQ7Ag2cC">General Topology</a>. <a href="Mineola%2C_N.Y." class="mw-redirect" title="Mineola, N.Y.">Mineola, N.Y.</a>: <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-43479-7</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/115240">115240</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.se16.info/hgb/nowhere.htm">Some nowhere dense sets with positive measure</a></li></ul></div><div><div class="zim-footer">

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