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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Formal definition</span> </div> </a> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationships_to_other_notions_of_dimension" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationships_to_other_notions_of_dimension"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Relationships to other notions of dimension</span> </div> </a> <ul id="toc-Relationships_to_other_notions_of_dimension-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 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class="mw-page-title-main">Lebesgue covering dimension</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://de.wikipedia.org/wiki/Lebesguesche_%C3%9Cberdeckungsdimension" title="Lebesguesche Überdeckungsdimension – German" lang="de" hreflang="de" data-title="Lebesguesche Überdeckungsdimension" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Dimensi%C3%B3n_de_recubrimiento_de_Lebesgue" title="Dimensión de recubrimiento de Lebesgue – Spanish" lang="es" hreflang="es" data-title="Dimensión de recubrimiento de Lebesgue" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Dimension_topologique" title="Dimension topologique – French" lang="fr" hreflang="fr" data-title="Dimension topologique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A5%B4%EB%B2%A0%EA%B7%B8_%EB%8D%AE%EA%B0%9C_%EC%B0%A8%EC%9B%90" title="르베그 덮개 차원 – Korean" lang="ko" hreflang="ko" data-title="르베그 덮개 차원" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Topolo%C5%A1ka_dimenzija" title="Topološka dimenzija – Croatian" lang="hr" hreflang="hr" data-title="Topološka dimenzija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Dimensione_topologica" title="Dimensione topologica – Italian" lang="it" hreflang="it" data-title="Dimensione topologica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Lefed%C3%A9si_dimenzi%C3%B3" title="Lefedési dimenzió – Hungarian" lang="hu" hreflang="hu" data-title="Lefedési dimenzió" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AB%E3%83%99%E3%83%BC%E3%82%B0%E8%A2%AB%E8%A6%86%E6%AC%A1%E5%85%83" title="ルベーグ被覆次元 – Japanese" lang="ja" hreflang="ja" data-title="ルベーグ被覆次元" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Dimens%C3%A3o_topol%C3%B3gica" title="Dimensão topológica – Portuguese" lang="pt" hreflang="pt" data-title="Dimensão topológica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BD%D0%BE%D1%81%D1%82%D1%8C_%D0%9B%D0%B5%D0%B1%D0%B5%D0%B3%D0%B0" title="Размерность Лебега – Russian" lang="ru" hreflang="ru" data-title="Размерность Лебега" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Topolo%C5%A1ka_dimenzija" title="Topološka dimenzija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Topološka dimenzija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D0%BC%D1%96%D1%80%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D0%9B%D0%B5%D0%B1%D0%B5%D0%B3%D0%B0" title="Розмірність Лебега – Ukrainian" lang="uk" hreflang="uk" data-title="Розмірність Лебега" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%8B%93%E6%92%B2%E7%B6%AD%E6%95%B8" title="拓撲維數 – Chinese" lang="zh" hreflang="zh" data-title="拓撲維數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Topological_dimension&redirect=no" class="mw-redirect" title="Topological dimension">Topological dimension</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Topologically invariant definition of the dimension of a space</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">April 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Lebesgue covering dimension</b> or <b>topological dimension</b> of a <a href="/wiki/Topological_space" title="Topological space">topological space</a> is one of several different ways of defining the <a href="/wiki/Dimension" title="Dimension">dimension</a> of the space in a <a href="/wiki/Topological_invariant" class="mw-redirect" title="Topological invariant">topologically invariant</a> way.<sup id="cite_ref-Lebesgue_1-0" class="reference"><a href="#cite_note-Lebesgue-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Duda_2-0" class="reference"><a href="#cite_note-Duda-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Informal_discussion">Informal discussion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=1" title="Edit section: Informal discussion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For ordinary <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean spaces</a>, the Lebesgue covering dimension is just the ordinary Euclidean dimension: zero for points, one for lines, two for planes, and so on. However, not all topological spaces have this kind of "obvious" <a href="/wiki/Dimension" title="Dimension">dimension</a>, and so a precise definition is needed in such cases. The definition proceeds by examining what happens when the space is covered by <a href="/wiki/Open_set" title="Open set">open sets</a>. </p><p>In general, a topological space <i>X</i> can be <a href="/wiki/Open_cover" class="mw-redirect" title="Open cover">covered by open sets</a>, in that one can find a collection of open sets such that <i>X</i> lies inside of their <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a>. The covering dimension is the smallest number <i>n</i> such that for every cover, there is a <a href="/wiki/Refinement_(topology)" class="mw-redirect" title="Refinement (topology)">refinement</a> in which every point in <i>X</i> lies in the <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of no more than <i>n</i> + 1 covering sets. This is the gist of the formal definition below. The goal of the definition is to provide a number (an <a href="/wiki/Integer" title="Integer">integer</a>) that describes the space, and does not change as the space is continuously deformed; that is, a number that is invariant under <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphisms</a>. </p><p>The general idea is illustrated in the diagrams below, which show a cover and refinements of a circle and a square. </p> <table> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Refinement_of_the_cover_of_a_circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Refinement_of_the_cover_of_a_circle.svg/220px-Refinement_of_the_cover_of_a_circle.svg.png" decoding="async" width="220" height="429" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Refinement_of_the_cover_of_a_circle.svg/330px-Refinement_of_the_cover_of_a_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Refinement_of_the_cover_of_a_circle.svg/440px-Refinement_of_the_cover_of_a_circle.svg.png 2x" data-file-width="378" data-file-height="737" /></a><figcaption>Refinement of the cover of a circle</figcaption></figure> </td> <td>The first image shows a refinement (on the bottom) of a colored cover (on the top) of a black circular line. Note how in the refinement, no point on the circle is contained in more than two sets, and also how the sets link to one another to form a "chain". </td></tr> <tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Refinement_on_a_planar_shape.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Refinement_on_a_planar_shape.svg/220px-Refinement_on_a_planar_shape.svg.png" decoding="async" width="220" height="429" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Refinement_on_a_planar_shape.svg/330px-Refinement_on_a_planar_shape.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Refinement_on_a_planar_shape.svg/440px-Refinement_on_a_planar_shape.svg.png 2x" data-file-width="378" data-file-height="737" /></a><figcaption>Refinement of the cover of a square</figcaption></figure> </td> <td>The top half of the second image shows a cover (colored) of a planar shape (dark), where all of the shape's points are contained in anywhere from one to all four of the cover's sets. The bottom illustrates that any attempt to refine said cover such that no point would be contained in more than <i>two</i> sets—ultimately fails at the intersection of set borders. Thus, a planar shape is not "webby": it cannot be covered with "chains", per se. Instead, it proves to be <i>thicker</i> in some sense. More rigorously put, its topological dimension must be greater than 1. </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=2" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Radford-stretcher-bond.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Radford-stretcher-bond.jpeg/220px-Radford-stretcher-bond.jpeg" decoding="async" width="220" height="66" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/8a/Radford-stretcher-bond.jpeg 1.5x" data-file-width="235" data-file-height="70" /></a><figcaption><a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Henri Lebesgue</a> used closed "bricks" to study covering dimension in 1921.<sup id="cite_ref-FOOTNOTELebesgue1921_3-0" class="reference"><a href="#cite_note-FOOTNOTELebesgue1921-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The first formal definition of covering dimension was given by <a href="/wiki/Eduard_%C4%8Cech" title="Eduard Čech">Eduard Čech</a>, based on an earlier result of <a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Henri Lebesgue</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>A modern definition is as follows. An <a href="/wiki/Open_cover" class="mw-redirect" title="Open cover">open cover</a> of a topological space <span class="texhtml mvar" style="font-style:italic;"><i>X</i></span> is a family of <a href="/wiki/Open_set" title="Open set">open sets</a> <span class="texhtml mvar" style="font-style:italic;"><i>U</i></span><sub><span class="texhtml mvar" style="font-style:italic;">α</span></sub> such that their union is the whole space, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cup _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54f1141599c6bb73f2a6f38c32934e9ce6fc389b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.834ex; height:2.343ex;" alt="{\displaystyle \cup _{\alpha }}"></span> <span class="texhtml mvar" style="font-style:italic;"><i>U</i></span><sub><span class="texhtml mvar" style="font-style:italic;">α</span></sub> = <span class="texhtml mvar" style="font-style:italic;"><i>X</i></span>. The <b>order</b> or <b>ply</b> of an open cover <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}"></span> = {<span class="texhtml mvar" style="font-style:italic;"><i>U</i></span><sub><span class="texhtml mvar" style="font-style:italic;">α</span></sub>} is the smallest number <span class="texhtml mvar" style="font-style:italic;"><i>m</i></span> (if it exists) for which each point of the space belongs to at most <span class="texhtml mvar" style="font-style:italic;"><i>m</i></span> open sets in the cover: in other words <span class="texhtml mvar" style="font-style:italic;">U</span><sub><span class="texhtml mvar" style="font-style:italic;">α</span><sub>1</sub></sub> ∩ ⋅⋅⋅ ∩ <span class="texhtml mvar" style="font-style:italic;">U</span><sub><span class="texhtml mvar" style="font-style:italic;">α</span><sub><span class="texhtml mvar" style="font-style:italic;"><i>m</i></span>+1</sub></sub> = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"></span> for <span class="texhtml mvar" style="font-style:italic;">α</span><sub>1</sub>, ..., <span class="texhtml mvar" style="font-style:italic;">α</span><sub><span class="texhtml mvar" style="font-style:italic;"><i>m</i></span>+1</sub> distinct. A <a href="/wiki/Refinement_(topology)" class="mw-redirect" title="Refinement (topology)">refinement</a> of an open cover <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}"></span> = {<span class="texhtml mvar" style="font-style:italic;"><i>U</i></span><sub><span class="texhtml mvar" style="font-style:italic;">α</span></sub>} is another open cover <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f939c87a07b7af23e09792e9edb2c7caebb18864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}"></span> = {<span class="texhtml mvar" style="font-style:italic;"><i>V</i></span><sub><span class="texhtml mvar" style="font-style:italic;">β</span></sub>}, such that each <span class="texhtml mvar" style="font-style:italic;"><i>V</i></span><sub><span class="texhtml mvar" style="font-style:italic;">β</span></sub> is contained in some <span class="texhtml mvar" style="font-style:italic;"><i>U</i></span><sub><span class="texhtml mvar" style="font-style:italic;">α</span></sub>. The <b>covering dimension</b> of a topological space <span class="texhtml mvar" style="font-style:italic;"><i>X</i></span> is defined to be the minimum value of <span class="texhtml mvar" style="font-style:italic;"><i>n</i></span> such that every finite open cover <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}"></span> of <i>X</i> has an open refinement <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f939c87a07b7af23e09792e9edb2c7caebb18864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}"></span> with order <span class="texhtml mvar" style="font-style:italic;"><i>n</i></span> + 1. The refinement <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f939c87a07b7af23e09792e9edb2c7caebb18864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}"></span> can always be chosen to be finite.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Thus, if <span class="texhtml mvar" style="font-style:italic;"><i>n</i></span> is finite, <span class="texhtml mvar" style="font-style:italic;">V</span><sub><span class="texhtml mvar" style="font-style:italic;">β</span><sub>1</sub></sub> ∩ ⋅⋅⋅ ∩ <span class="texhtml mvar" style="font-style:italic;">V</span><sub><span class="texhtml mvar" style="font-style:italic;">β</span><sub><span class="texhtml mvar" style="font-style:italic;"><i>n</i></span>+2</sub></sub> = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"></span> for <span class="texhtml mvar" style="font-style:italic;">β</span><sub>1</sub>, ..., <span class="texhtml mvar" style="font-style:italic;">β</span><sub><span class="texhtml mvar" style="font-style:italic;"><i>n</i></span>+2</sub> distinct. If no such minimal <span class="texhtml mvar" style="font-style:italic;"><i>n</i></span> exists, the space is said to have infinite covering dimension. </p><p>As a special case, a non-empty topological space is <a href="/wiki/Zero-dimensional_space" title="Zero-dimensional space">zero-dimensional</a> with respect to the covering dimension if every open cover of the space has a refinement consisting of <a href="/wiki/Disjoint_set" class="mw-redirect" title="Disjoint set">disjoint</a> open sets, meaning any point in the space is contained in exactly one open set of this refinement. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The empty set has covering dimension -1: for any open cover of the empty set, each point of the empty set is not contained in any element of the cover, so the order of any open cover is 0. </p><p>Any given open cover of the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> will have a refinement consisting of a collection of <a href="/wiki/Open_(topology)" class="mw-redirect" title="Open (topology)">open</a> arcs. The circle has dimension one, by this definition, because any such cover can be further refined to the stage where a given point <i>x</i> of the circle is contained in <i>at most</i> two open arcs. That is, whatever collection of arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle but with simple overlaps. </p><p>Similarly, any open cover of the <a href="/wiki/Unit_disk" title="Unit disk">unit disk</a> in the two-dimensional <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a> can be refined so that any point of the disk is contained in no more than three open sets, while two are in general not sufficient. The covering dimension of the disk is thus two. </p><p>More generally, the <i>n</i>-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8fa8586d428ff5706c6d0a00a7939950fad89b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {E} ^{n}}"></span> has covering dimension <i>n</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">Homeomorphic</a> spaces have the same covering dimension. That is, the covering dimension is a <a href="/wiki/Topological_invariant" class="mw-redirect" title="Topological invariant">topological invariant</a>.</li> <li>The covering dimension of a normal space <i>X</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61fbd99c9375a5553cc47f02b7da90cb8becd4ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.848ex; height:2.176ex;" alt="{\displaystyle \leq n}"></span> if and only if for any <a href="/wiki/Closed_subset" class="mw-redirect" title="Closed subset">closed subset</a> <i>A</i> of <i>X</i>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\rightarrow S^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\rightarrow S^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49019e8db765c799c637da709ad79270723747ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.313ex; height:2.676ex;" alt="{\displaystyle f:A\rightarrow S^{n}}"></span> is continuous, then there is an extension of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:X\rightarrow S^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:X\rightarrow S^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faad65f06630bb6b2dd570accc64fb6110dde801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.387ex; height:2.676ex;" alt="{\displaystyle g:X\rightarrow S^{n}}"></span>. Here, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee006452a59bf1eb29983b4412348b66517a2d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.343ex;" alt="{\displaystyle S^{n}}"></span> is the <a href="/wiki/N-sphere" title="N-sphere"><i>n</i>-dimensional sphere</a>.</li> <li><b>Ostrand's theorem on colored dimension.</b> If <span class="texhtml mvar" style="font-style:italic;">X</span> is a normal topological space and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}"></span> = {<span class="texhtml mvar" style="font-style:italic;"><i>U</i></span><sub><span class="texhtml mvar" style="font-style:italic;">α</span></sub>} is a locally finite cover of <span class="texhtml mvar" style="font-style:italic;"><i>X</i></span> of order ≤ <span class="texhtml mvar" style="font-style:italic;"><i>n</i></span> + 1, then, for each 1 ≤ <span class="texhtml mvar" style="font-style:italic;"><i>i</i></span> ≤ <span class="texhtml mvar" style="font-style:italic;"><i>n</i></span> + 1, there exists a family of pairwise disjoint open sets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f939c87a07b7af23e09792e9edb2c7caebb18864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}"></span><sub><span class="texhtml mvar" style="font-style:italic;"><i>i</i></span></sub> = {<span class="texhtml mvar" style="font-style:italic;"><i>V</i></span><sub><span class="texhtml mvar" style="font-style:italic;"><i>i</i></span>,<span class="texhtml mvar" style="font-style:italic;">α</span></sub>} shrinking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}"></span>, i.e. <span class="texhtml mvar" style="font-style:italic;"><i>V</i></span><sub><span class="texhtml mvar" style="font-style:italic;"><i>i</i></span>,<span class="texhtml mvar" style="font-style:italic;">α</span></sub> ⊆ <span class="texhtml mvar" style="font-style:italic;"><i>U</i></span><sub><span class="texhtml mvar" style="font-style:italic;">α</span></sub>, and together covering <span class="texhtml mvar" style="font-style:italic;"><i>X</i></span>.<sup id="cite_ref-FOOTNOTEOstrand1971_6-0" class="reference"><a href="#cite_note-FOOTNOTEOstrand1971-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Relationships_to_other_notions_of_dimension">Relationships to other notions of dimension</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=5" title="Edit section: Relationships to other notions of dimension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>For a paracompact space <span class="texhtml mvar" style="font-style:italic;"><i>X</i></span>, the covering dimension can be equivalently defined as the minimum value of <span class="texhtml mvar" style="font-style:italic;"><i>n</i></span>, such that every open cover <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}"></span> of <span class="texhtml mvar" style="font-style:italic;"><i>X</i></span> (of any size) has an open refinement <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f939c87a07b7af23e09792e9edb2c7caebb18864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}"></span> with order <span class="texhtml mvar" style="font-style:italic;"><i>n</i></span> + 1.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In particular, this holds for all metric spaces.</li> <li><b>Lebesgue covering theorem.</b> The Lebesgue covering dimension coincides with the <a href="/w/index.php?title=Affine_dimension&action=edit&redlink=1" class="new" title="Affine dimension (page does not exist)">affine dimension</a> of a finite <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a>.</li> <li>The covering dimension of a <a href="/wiki/Normal_space" title="Normal space">normal space</a> is less than or equal to the large <a href="/wiki/Inductive_dimension" title="Inductive dimension">inductive dimension</a>.</li> <li>The covering dimension of a <a href="/wiki/Paracompact_space" title="Paracompact space">paracompact</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> is greater or equal to its <a href="/wiki/Cohomological_dimension" title="Cohomological dimension">cohomological dimension</a> (in the sense of <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaves</a>),<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> that is, one has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{i}(X,A)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{i}(X,A)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42a3701cb782aad7b912502bd1478f849bab38bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.73ex; height:3.176ex;" alt="{\displaystyle H^{i}(X,A)=0}"></span> for every sheaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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}"></span> of abelian groups on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> and every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> larger than the covering dimension of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span>.</li> <li>In a <a href="/wiki/Metric_space" title="Metric space">metric space</a>, one can strengthen the notion of the multiplicity of a cover: a cover has <i><span class="texhtml mvar" style="font-style:italic;">r</span>-multiplicity</i> <span class="texhtml"><i>n</i> + 1</span> if every <span class="texhtml mvar" style="font-style:italic;">r</span>-ball intersects with at most <span class="texhtml mvar" style="font-style:italic;"><i>n</i> + 1</span> sets in the cover. This idea leads to the definitions of the <a href="/wiki/Asymptotic_dimension" title="Asymptotic dimension">asymptotic dimension</a> and <a href="/wiki/Assouad%E2%80%93Nagata_dimension" title="Assouad–Nagata dimension">Assouad–Nagata dimension</a> of a space: a space with asymptotic dimension <span class="texhtml mvar" style="font-style:italic;">n</span> is <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional "at large scales", and a space with Assouad–Nagata dimension <span class="texhtml mvar" style="font-style:italic;">n</span> is <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional "at every scale".</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Carath%C3%A9odory%27s_extension_theorem" title="Carathéodory's extension theorem">Carathéodory's extension theorem</a></li> <li><a href="/wiki/Geometric_set_cover_problem" title="Geometric set cover problem">Geometric set cover problem</a></li> <li><a href="/wiki/Dimension_theory" class="mw-redirect" title="Dimension theory">Dimension theory</a></li> <li><a href="/wiki/Metacompact_space" title="Metacompact space">Metacompact space</a></li> <li><a href="/wiki/Point-finite_collection" title="Point-finite collection">Point-finite collection</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Lebesgue-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lebesgue_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLebesgue1921" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Lebesgue, Henri</a> (1921). <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/fm/fm2/fm2130.pdf">"Sur les correspondances entre les points de deux espaces"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Fundamenta_Mathematicae" title="Fundamenta Mathematicae">Fundamenta Mathematicae</a></i> (in French). <b>2</b>: 256–285. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Ffm-2-1-256-285">10.4064/fm-2-1-256-285</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fundamenta+Mathematicae&rft.atitle=Sur+les+correspondances+entre+les+points+de+deux+espaces&rft.volume=2&rft.pages=256-285&rft.date=1921&rft_id=info%3Adoi%2F10.4064%2Ffm-2-1-256-285&rft.aulast=Lebesgue&rft.aufirst=Henri&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Ffm%2Ffm2%2Ffm2130.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></span> </li> <li id="cite_note-Duda-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Duda_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuda1979" class="citation journal cs1">Duda, R. (1979). <a rel="nofollow" class="external text" href="https://www.impan.pl/en/publishing-house/journals-and-series/colloquium-mathematicum/all/42/1/102445/the-origins-of-the-concept-of-dimension">"The origins of the concept of dimension"</a>. <i>Colloquium Mathematicum</i>. <b>42</b>: 95–110. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Fcm-42-1-95-110">10.4064/cm-42-1-95-110</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0567548">0567548</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Colloquium+Mathematicum&rft.atitle=The+origins+of+the+concept+of+dimension&rft.volume=42&rft.pages=95-110&rft.date=1979&rft_id=info%3Adoi%2F10.4064%2Fcm-42-1-95-110&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0567548%23id-name%3DMR&rft.aulast=Duda&rft.aufirst=R.&rft_id=https%3A%2F%2Fwww.impan.pl%2Fen%2Fpublishing-house%2Fjournals-and-series%2Fcolloquium-mathematicum%2Fall%2F42%2F1%2F102445%2Fthe-origins-of-the-concept-of-dimension&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTELebesgue1921-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELebesgue1921_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLebesgue1921">Lebesgue 1921</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKuperberg1995" class="citation cs2"><a href="/wiki/Krystyna_Kuperberg" title="Krystyna Kuperberg">Kuperberg, Krystyna</a>, ed. (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6EICfJrepKQC&pg=PR23"><i>Collected Works of Witold Hurewicz</i></a>, American Mathematical Society, Collected works series, vol. 4, American Mathematical Society, p. xxiii, footnote 3, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821800119" title="Special:BookSources/9780821800119"><bdi>9780821800119</bdi></a>, <q>Lebesgue's discovery led later to the introduction by E. Čech of the covering dimension</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Collected+Works+of+Witold+Hurewicz&rft.series=American+Mathematical+Society%2C+Collected+works+series&rft.pages=p.-xxiii%2C+footnote-3&rft.pub=American+Mathematical+Society&rft.date=1995&rft.isbn=9780821800119&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6EICfJrepKQC%26pg%3DPR23&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Proposition 1.6.9 of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEngelking1978" class="citation book cs1">Engelking, Ryszard (1978). <a rel="nofollow" class="external text" href="https://www.maths.ed.ac.uk/~v1ranick/papers/engelking.pdf"><i>Dimension theory</i></a> <span class="cs1-format">(PDF)</span>. North-Holland Mathematical Library. Vol. 19. Amsterdam-Oxford-New York: North-Holland. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-85176-3" title="Special:BookSources/0-444-85176-3"><bdi>0-444-85176-3</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0482697">0482697</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dimension+theory&rft.place=Amsterdam-Oxford-New+York&rft.series=North-Holland+Mathematical+Library&rft.pub=North-Holland&rft.date=1978&rft.isbn=0-444-85176-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0482697%23id-name%3DMR&rft.aulast=Engelking&rft.aufirst=Ryszard&rft_id=https%3A%2F%2Fwww.maths.ed.ac.uk%2F~v1ranick%2Fpapers%2Fengelking.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEOstrand1971-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEOstrand1971_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFOstrand1971">Ostrand 1971</a>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Proposition 3.2.2 of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEngelking1978" class="citation book cs1">Engelking, Ryszard (1978). <a rel="nofollow" class="external text" href="https://www.maths.ed.ac.uk/~v1ranick/papers/engelking.pdf"><i>Dimension theory</i></a> <span class="cs1-format">(PDF)</span>. North-Holland Mathematical Library. Vol. 19. Amsterdam-Oxford-New York: North-Holland. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-85176-3" title="Special:BookSources/0-444-85176-3"><bdi>0-444-85176-3</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0482697">0482697</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dimension+theory&rft.place=Amsterdam-Oxford-New+York&rft.series=North-Holland+Mathematical+Library&rft.pub=North-Holland&rft.date=1978&rft.isbn=0-444-85176-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0482697%23id-name%3DMR&rft.aulast=Engelking&rft.aufirst=Ryszard&rft_id=https%3A%2F%2Fwww.maths.ed.ac.uk%2F~v1ranick%2Fpapers%2Fengelking.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Godement 1973, II.5.12, p. 236</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdgar2008" class="citation book cs1">Edgar, Gerald A. (2008). "Topological Dimension". <i>Measure, topology, and fractal geometry</i>. Undergraduate Texts in Mathematics (Second ed.). <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. 85–114. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-74748-4" title="Special:BookSources/978-0-387-74748-4"><bdi>978-0-387-74748-4</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2356043">2356043</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Topological+Dimension&rft.btitle=Measure%2C+topology%2C+and+fractal+geometry&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=85-114&rft.edition=Second&rft.pub=Springer-Verlag&rft.date=2008&rft.isbn=978-0-387-74748-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2356043%23id-name%3DMR&rft.aulast=Edgar&rft.aufirst=Gerald+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEngelking1978" class="citation book cs1">Engelking, Ryszard (1978). <a rel="nofollow" class="external text" href="https://www.maths.ed.ac.uk/~v1ranick/papers/engelking.pdf"><i>Dimension theory</i></a> <span class="cs1-format">(PDF)</span>. North-Holland Mathematical Library. Vol. 19. Amsterdam-Oxford-New York: North-Holland. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-85176-3" title="Special:BookSources/0-444-85176-3"><bdi>0-444-85176-3</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0482697">0482697</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dimension+theory&rft.place=Amsterdam-Oxford-New+York&rft.series=North-Holland+Mathematical+Library&rft.pub=North-Holland&rft.date=1978&rft.isbn=0-444-85176-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0482697%23id-name%3DMR&rft.aulast=Engelking&rft.aufirst=Ryszard&rft_id=https%3A%2F%2Fwww.maths.ed.ac.uk%2F~v1ranick%2Fpapers%2Fengelking.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGodement1958" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Roger_Godement" title="Roger Godement">Godement, Roger</a> (1958). <i>Topologie algébrique et théorie des faisceaux</i>. Publications de l'Institut de Mathématique de l'Université de Strasbourg (in French). Vol. III. Paris: Hermann. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0102797">0102797</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topologie+alg%C3%A9brique+et+th%C3%A9orie+des+faisceaux&rft.place=Paris&rft.series=Publications+de+l%27Institut+de+Math%C3%A9matique+de+l%27Universit%C3%A9+de+Strasbourg&rft.pub=Hermann&rft.date=1958&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0102797%23id-name%3DMR&rft.aulast=Godement&rft.aufirst=Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHurewiczWallman1941" class="citation book cs1">Hurewicz, Witold; Wallman, Henry (1941). <i>Dimension Theory</i>. Princeton Mathematical Series. Vol. 4. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0006493">0006493</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dimension+Theory&rft.series=Princeton+Mathematical+Series&rft.pub=Princeton+University+Press&rft.date=1941&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0006493%23id-name%3DMR&rft.aulast=Hurewicz&rft.aufirst=Witold&rft.au=Wallman%2C+Henry&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMunkres2000" class="citation book cs1"><a href="/wiki/James_Munkres" title="James Munkres">Munkres, James R.</a> (2000). <i>Topology</i> (2nd ed.). Prentice-Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-181629-2" title="Special:BookSources/0-13-181629-2"><bdi>0-13-181629-2</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3728284">3728284</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology&rft.edition=2nd&rft.pub=Prentice-Hall&rft.date=2000&rft.isbn=0-13-181629-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3728284%23id-name%3DMR&rft.aulast=Munkres&rft.aufirst=James+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOstrand1971" class="citation journal cs1">Ostrand, Phillip A. (1971). "Covering dimension in general spaces". <i>General Topology and Appl</i>. <b>1</b> (3): 209–221. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0288741">0288741</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=General+Topology+and+Appl.&rft.atitle=Covering+dimension+in+general+spaces&rft.volume=1&rft.issue=3&rft.pages=209-221&rft.date=1971&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0288741%23id-name%3DMR&rft.aulast=Ostrand&rft.aufirst=Phillip+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Historical">Historical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=10" title="Edit section: Historical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Karl_Menger" title="Karl Menger">Karl Menger</a>, <i>General Spaces and Cartesian Spaces</i>, (1926) Communications to the Amsterdam Academy of Sciences. English translation reprinted in <i>Classics on Fractals</i>, Gerald A.Edgar, editor, Addison-Wesley (1993) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-58701-7" title="Special:BookSources/0-201-58701-7">0-201-58701-7</a></li> <li><a href="/wiki/Karl_Menger" title="Karl Menger">Karl Menger</a>, <i>Dimensionstheorie</i>, (1928) B.G Teubner Publishers, Leipzig.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Modern">Modern</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=11" title="Edit section: Modern"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPears1975" class="citation book cs1">Pears, Alan R. (1975). <a rel="nofollow" class="external text" href="https://archive.org/details/dimensiontheoryo0000pear"><i>Dimension Theory of General Spaces</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-20515-8" title="Special:BookSources/0-521-20515-8"><bdi>0-521-20515-8</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0394604">0394604</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dimension+Theory+of+General+Spaces&rft.pub=Cambridge+University+Press&rft.date=1975&rft.isbn=0-521-20515-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0394604%23id-name%3DMR&rft.aulast=Pears&rft.aufirst=Alan+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdimensiontheoryo0000pear&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></li> <li>V. V. Fedorchuk, <i>The Fundamentals of Dimension Theory</i>, appearing in <i>Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I</i>, (1993) A. V. Arkhangel'skii and <a href="/wiki/L._S._Pontryagin" class="mw-redirect" title="L. S. Pontryagin">L. S. Pontryagin</a> (Eds.), Springer-Verlag, Berlin <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-18178-4" title="Special:BookSources/3-540-18178-4">3-540-18178-4</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lebesgue_covering_dimension&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Lebesgue_dimension">"Lebesgue dimension"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Lebesgue+dimension&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DLebesgue_dimension&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALebesgue+covering+dimension" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output 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href="/wiki/Template_talk:Dimension_topics" title="Template talk:Dimension topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Dimension_topics" title="Special:EditPage/Template:Dimension topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Dimension" style="font-size:114%;margin:0 4em"><a href="/wiki/Dimension" title="Dimension">Dimension</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dimensional spaces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">Vector space</a></li> <li><a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a></li> <li><a href="/wiki/Affine_space" title="Affine space">Affine space</a></li> <li><a href="/wiki/Projective_space" title="Projective space">Projective space</a></li> <li><a href="/wiki/Free_module" title="Free module">Free module</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">Algebraic variety</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Tesseract.gif" class="mw-file-description" title="Animated tesseract"><img alt="Animated tesseract" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tesseract.gif/75px-Tesseract.gif" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tesseract.gif/113px-Tesseract.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tesseract.gif/150px-Tesseract.gif 2x" data-file-width="256" data-file-height="256" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other dimensions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Krull_dimension" title="Krull dimension">Krull</a></li> <li><a class="mw-selflink selflink">Lebesgue covering</a></li> <li><a href="/wiki/Inductive_dimension" title="Inductive dimension">Inductive</a></li> <li><a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff</a></li> <li><a href="/wiki/Minkowski%E2%80%93Bouligand_dimension" title="Minkowski–Bouligand dimension">Minkowski</a></li> <li><a href="/wiki/Fractal_dimension" title="Fractal dimension">Fractal</a></li> <li><a href="/wiki/Degrees_of_freedom" title="Degrees of freedom">Degrees of freedom</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polytope" title="Polytope">Polytopes</a> and <a href="/wiki/Shape" title="Shape">shapes</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hyperplane" title="Hyperplane">Hyperplane</a></li> <li><a href="/wiki/Hypersurface" title="Hypersurface">Hypersurface</a></li> <li><a href="/wiki/Hypercube" title="Hypercube">Hypercube</a></li> <li><a href="/wiki/Hyperrectangle" title="Hyperrectangle">Hyperrectangle</a></li> <li><a href="/wiki/Demihypercube" title="Demihypercube">Demihypercube</a></li> <li><a href="/wiki/N-sphere" title="N-sphere">Hypersphere</a></li> <li><a href="/wiki/Cross-polytope" title="Cross-polytope">Cross-polytope</a></li> <li><a href="/wiki/Simplex" title="Simplex">Simplex</a></li> <li><a href="/wiki/Hyperpyramid" title="Hyperpyramid">Hyperpyramid</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Number systems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hypercomplex_number" title="Hypercomplex number">Hypercomplex numbers</a></li> <li><a href="/wiki/Cayley%E2%80%93Dickson_construction" title="Cayley–Dickson construction">Cayley–Dickson construction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dimensions by number</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zero-dimensional_space" title="Zero-dimensional space">Zero</a></li> <li><a href="/wiki/One-dimensional_space" title="One-dimensional space">One</a></li> <li><a href="/wiki/Two-dimensional_space" title="Two-dimensional space">Two</a></li> <li><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Three</a></li> <li><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">Four</a></li> <li><a href="/wiki/Five-dimensional_space" title="Five-dimensional space">Five</a></li> <li><a href="/wiki/Six-dimensional_space" title="Six-dimensional space">Six</a></li> <li><a href="/wiki/Seven-dimensional_space" title="Seven-dimensional space">Seven</a></li> <li><a href="/wiki/Eight-dimensional_space" title="Eight-dimensional space">Eight</a></li> <li><a href="/wiki/Dimension" title="Dimension"><i>n</i>-dimensions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hyperspace" title="Hyperspace">Hyperspace</a></li> <li><a href="/wiki/Codimension" title="Codimension">Codimension</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><b><a href="/wiki/Category:Dimension" title="Category:Dimension">Category</a></b></div></td></tr></tbody></table></div>    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