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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Se connecter</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages pour les contributeurs déconnectés <a href="/wiki/Aide:Premiers_pas" aria-label="En savoir plus sur la contribution"><span>en savoir plus</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Mes_contributions" title="Une liste des modifications effectuées depuis cette adresse IP [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Mes_discussions" title="La page de discussion pour les contributions depuis cette adresse IP [n]" accesskey="n"><span>Discussion</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Sommaire" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Sommaire</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">masquer</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Début</div> </a> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Voir aussi</span> </div> </a> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Sommaire" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Basculer la table des matières" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Basculer la table des matières</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Propriété locale</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="Aller à un article dans une autre langue. Disponible en 7 langues." > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-7" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">7 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lokal_(Topologie)" title="Lokal (Topologie) – allemand" lang="de" hreflang="de" data-title="Lokal (Topologie)" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Local_property" title="Local property – anglais" lang="en" hreflang="en" data-title="Local property" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Localmente" title="Localmente – espagnol" lang="es" hreflang="es" data-title="Localmente" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/W%C5%82asno%C5%9B%C4%87_lokalna" title="Własność lokalna – polonais" lang="pl" hreflang="pl" data-title="Własność lokalna" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Localmente_(topologia)" title="Localmente (topologia) – portugais" lang="pt" hreflang="pt" data-title="Localmente (topologia)" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B1%80%E9%83%A8%E6%80%A7%E8%B4%A8" title="局部性质 – chinois" lang="zh" hreflang="zh" data-title="局部性质" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%B1%80%E9%83%A8" title="局部 – chinois littéraire" lang="lzh" hreflang="lzh" data-title="局部" data-language-autonym="文言" data-language-local-name="chinois littéraire" class="interlanguage-link-target"><span>文言</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1868414#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaces de noms"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Propri%C3%A9t%C3%A9_locale" title="Voir le contenu de la page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussion:Propri%C3%A9t%C3%A9_locale" rel="discussion" title="Discussion au sujet de cette page de contenu [t]" accesskey="t"><span>Discussion</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Modifier la variante de langue" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">français</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Affichages"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Propri%C3%A9t%C3%A9_locale"><span>Lire</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Propri%C3%A9t%C3%A9_locale&veaction=edit" title="Modifier cette page [v]" accesskey="v"><span>Modifier</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Propri%C3%A9t%C3%A9_locale&action=edit" title="Modifier le wikicode de cette page [e]" accesskey="e"><span>Modifier le code</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Propri%C3%A9t%C3%A9_locale&action=history" title="Historique des versions de cette page [h]" accesskey="h"><span>Voir l’historique</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Outils de la page"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Outils" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Outils</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Outils</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">masquer</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Plus d’options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Propri%C3%A9t%C3%A9_locale"><span>Lire</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Propri%C3%A9t%C3%A9_locale&veaction=edit" title="Modifier cette page [v]" accesskey="v"><span>Modifier</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Propri%C3%A9t%C3%A9_locale&action=edit" title="Modifier le wikicode de cette page [e]" accesskey="e"><span>Modifier le code</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Propri%C3%A9t%C3%A9_locale&action=history"><span>Voir l’historique</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Général </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_li%C3%A9es/Propri%C3%A9t%C3%A9_locale" title="Liste des pages liées qui pointent sur celle-ci [j]" accesskey="j"><span>Pages liées</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Suivi_des_liens/Propri%C3%A9t%C3%A9_locale" rel="nofollow" title="Liste des modifications récentes des pages appelées par celle-ci [k]" accesskey="k"><span>Suivi des pages liées</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Aide:Importer_un_fichier" title="Téléverser des fichiers [u]" accesskey="u"><span>Téléverser un fichier</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_sp%C3%A9ciales" title="Liste de toutes les pages spéciales [q]" accesskey="q"><span>Pages spéciales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Propri%C3%A9t%C3%A9_locale&oldid=219103276" title="Adresse permanente de cette version de cette page"><span>Lien permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Propri%C3%A9t%C3%A9_locale&action=info" title="Davantage d’informations sur cette page"><span>Informations sur la page</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:Citer&page=Propri%C3%A9t%C3%A9_locale&id=219103276&wpFormIdentifier=titleform" title="Informations sur la manière de citer cette page"><span>Citer cette page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:UrlShortener&url=https%3A%2F%2Ffr.wikipedia.org%2Fwiki%2FPropri%25C3%25A9t%25C3%25A9_locale"><span>Obtenir l'URL raccourcie</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:QrCode&url=https%3A%2F%2Ffr.wikipedia.org%2Fwiki%2FPropri%25C3%25A9t%25C3%25A9_locale"><span>Télécharger le code QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimer / exporter </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:Livre&bookcmd=book_creator&referer=Propri%C3%A9t%C3%A9+locale"><span>Créer un livre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:DownloadAsPdf&page=Propri%C3%A9t%C3%A9_locale&action=show-download-screen"><span>Télécharger comme PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Propri%C3%A9t%C3%A9_locale&printable=yes" title="Version imprimable de cette page [p]" accesskey="p"><span>Version imprimable</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dans d’autres projets </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1868414" title="Lien vers l’élément dans le dépôt de données connecté [g]" accesskey="g"><span>Élément Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Outils de la page"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apparence"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apparence</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Un article de Wikipédia, l'encyclopédie libre.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><div class="bandeau-container metadata bandeau-article bandeau-niveau-ebauche"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Fichier:KleinBottle-01.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/KleinBottle-01.svg/39px-KleinBottle-01.svg.png" decoding="async" width="39" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/KleinBottle-01.svg/59px-KleinBottle-01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/KleinBottle-01.svg/78px-KleinBottle-01.svg.png 2x" data-file-width="668" data-file-height="600" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p><strong class="bandeau-titre">Cet article est une <a href="/wiki/Aide:%C3%89bauche" title="Aide:Ébauche">ébauche</a> concernant la <a href="/wiki/Topologie" title="Topologie">topologie</a>.</strong> </p><p>Vous pouvez partager vos connaissances en l’améliorant (<b><a href="/wiki/Aide:Comment_modifier_une_page" title="Aide:Comment modifier une page">comment ?</a></b>) selon les recommandations des <a href="/wiki/Projet:Accueil" title="Projet:Accueil">projets correspondants</a>. </p> </div></div> <p>On dit d'une certaine propriété <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématique</a> qu'elle est <b>localement</b> vérifiée en un point d'un <a href="/wiki/Espace_topologique" title="Espace topologique">espace topologique</a> s'il existe un <a href="/wiki/Glossaire_topologique#S" class="mw-redirect" title="Glossaire topologique">système fondamental de voisinages</a> de ce point sur lequel la propriété est vraie. </p><p>On dit d'une certaine propriété mathématique qu'elle est <b>localement</b> vérifiée si elle est localement vérifiée en tout point de l'espace topologique considéré. </p><p>Cette notion se retrouve dans tous les domaines des <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a> qui utilisent la <a href="/wiki/Topologie" title="Topologie">topologie</a>, en particulier en <a href="/wiki/Analyse_(math%C3%A9matiques)" title="Analyse (mathématiques)">analyse</a>. </p><p>Souvent, il suffit que la propriété soit vraie pour un voisinage du point pour qu'elle soit vraie localement en ce point, par exemple : </p> <ul><li>On dit qu'une fonction <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:X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f:X\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c220d37b636b629696c6b28c55b81d15df8e255e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.069ex; height:2.509ex;" alt="{\displaystyle \ f:X\to \mathbb {R} }"></span> définie sur un espace topologique <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"> <mtext> </mtext> <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/a96dd62d4cca19aa212ae1216891f4388ca4be24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.561ex; height:2.176ex;" alt="{\displaystyle \ X}"></span> admet en un point <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"> <mtext> </mtext> <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/a14b4bc0b189c2d21060cd9ed1516a1f7e707c85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.81ex; height:1.676ex;" alt="{\displaystyle \ a}"></span> de <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"> <mtext> </mtext> <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/a96dd62d4cca19aa212ae1216891f4388ca4be24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.561ex; height:2.176ex;" alt="{\displaystyle \ X}"></span> un <a href="/wiki/Plus_grand_%C3%A9l%C3%A9ment" class="mw-redirect" title="Plus grand élément"> maximum local</a> s'il existe un voisinage <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 \ V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/480929376c7e702582861e5967e68f4a539fe7a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.368ex; height:2.176ex;" alt="{\displaystyle \ V}"></span> de <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"> <mtext> </mtext> <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/a14b4bc0b189c2d21060cd9ed1516a1f7e707c85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.81ex; height:1.676ex;" alt="{\displaystyle \ a}"></span> tel que <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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93fdea07afca1ed0f9fd25df005ce4693f256ca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.898ex; height:2.843ex;" alt="{\displaystyle \ f(a)}"></span> soit la plus grande valeur de <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"> <mtext> </mtext> <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/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> sur <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 \ V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/480929376c7e702582861e5967e68f4a539fe7a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.368ex; height:2.176ex;" alt="{\displaystyle \ V}"></span> ;</li> <li>On dit qu'un espace topologique est <a href="/wiki/Espace_localement_compact" title="Espace localement compact">localement compact</a> s'il est <a href="/wiki/Espace_s%C3%A9par%C3%A9" title="Espace séparé">séparé</a> et si chacun de ses points possède un voisinage <a href="/wiki/Compacit%C3%A9_(math%C3%A9matiques)" title="Compacité (mathématiques)">compact</a>.</li></ul> <p>Cependant, c'est faux en général, par exemple : </p> <ul><li>On ne peut pas dire que la fonction <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\mapsto {\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x\mapsto {\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ea79fbf6008589719542c970a57be45661dc77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.69ex; height:5.176ex;" alt="{\displaystyle \ x\mapsto {\frac {1}{x}}}"></span> est localement non bornée au point <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=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc82dc93cd3017c8e5e7989316a6475fbd07e5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.171ex; height:2.176ex;" alt="{\displaystyle \ x=1}"></span> sous prétexte qu'elle est non bornée sur <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 \ ]0,2[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">[</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ ]0,2[}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62938da5f5e0fab00729e0b508288dfa9f497657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.233ex; height:2.843ex;" alt="{\displaystyle \ ]0,2[}"></span> (qui est un voisinage du point considéré) ;</li> <li>Il existe des espaces connexes qui ne sont pas localement connexes en un point, on peut par exemple prendre l'adhérence du graphe de <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\mapsto \sin {\frac {1}{x}},x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>,</mo> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x\mapsto \sin {\frac {1}{x}},x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc6ecfc9cdedf7be447bbfb763050fbfbfe3ccf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.557ex; height:5.176ex;" alt="{\displaystyle \ x\mapsto \sin {\frac {1}{x}},x>0}"></span>, dans <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 {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/708ca61729506944e7add676e3a50a4b875129a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.313ex; height:2.676ex;" alt="{\displaystyle \ \mathbb {R} ^{2}}"></span>, et le point <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 \ (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b46beafa1c4140bc5712e33af2140ab601d449e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.749ex; height:2.843ex;" alt="{\displaystyle \ (0,1)}"></span>. Pourtant, l'espace tout entier est un voisinage connexe de ce point.</li></ul> <p>Cette expression intervient également en <a href="/wiki/Th%C3%A9orie_des_groupes" title="Théorie des groupes">théorie des groupes</a> : on dit qu'un <a href="/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">groupe</a> vérifie localement une propriété si tous ses <a href="/wiki/Sous-groupe" title="Sous-groupe">sous-groupes</a> <a href="/wiki/Groupe_de_type_fini" class="mw-redirect" title="Groupe de type fini">de type fini</a> la vérifient. Par exemple, un groupe est <a href="/w/index.php?title=Localement_nilpotent&action=edit&redlink=1" class="new" title="Localement nilpotent (page inexistante)">localement nilpotent</a> <a href="https://en.wikipedia.org/wiki/Locally_nilpotent" class="extiw" title="en:Locally nilpotent"><span class="indicateur-langue" title="Article en anglais : « Locally nilpotent »">(en)</span></a> si tous ses sous-groupes de type fini sont <a href="/wiki/Groupe_nilpotent" title="Groupe nilpotent">nilpotents</a> ; il est <a href="/w/index.php?title=Groupe_localement_fini&action=edit&redlink=1" class="new" title="Groupe localement fini (page inexistante)">localement fini</a> <a href="https://en.wikipedia.org/wiki/Locally_finite_group" class="extiw" title="en:Locally finite group"><span class="indicateur-langue" title="Article en anglais : « Locally finite group »">(en)</span></a> si tous ses sous-groupes de type fini sont <a href="/wiki/Groupe_fini" title="Groupe fini">finis</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propri%C3%A9t%C3%A9_locale&veaction=edit&section=1" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propri%C3%A9t%C3%A9_locale&action=edit&section=1" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Espace_localement_compact" title="Espace localement compact">Espace localement compact</a></li> <li><a href="/wiki/Espace_localement_connexe" title="Espace localement connexe">Espace localement connexe</a></li> <li><a href="/wiki/Espace_localement_connexe_par_arcs" title="Espace localement connexe par arcs">Espace localement connexe par arcs</a></li> <li><a href="/wiki/Espace_localement_simplement_connexe" title="Espace localement simplement connexe">Espace localement simplement connexe</a></li> <li><a href="/wiki/Espace_s%C3%A9par%C3%A9#Espace_localement_séparé" title="Espace séparé">Espace localement séparé</a></li> <li><a href="/w/index.php?title=Fonction_localement_born%C3%A9e&action=edit&redlink=1" class="new" title="Fonction localement bornée (page inexistante)">Fonction localement bornée</a> <a href="https://en.wikipedia.org/wiki/Local_boundedness" class="extiw" title="en:Local boundedness"><span class="indicateur-langue" title="Article en anglais : « Local boundedness »">(en)</span></a></li> <li><a href="/wiki/Hom%C3%A9omorphisme#Définitions_associées" title="Homéomorphisme">Homéomorphisme local</a>, <a href="/wiki/Diff%C3%A9omorphisme#Difféomorphisme_local" title="Difféomorphisme">Difféomorphisme local</a></li> <li><a href="/wiki/Glossaire_topologique#L" class="mw-redirect" title="Glossaire topologique">Famille localement finie de parties</a></li> <li><a href="/wiki/Mesure_localement_finie" title="Mesure localement finie">Mesure localement finie</a></li></ul> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail des mathématiques"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/24px-Racine_carr%C3%A9e_bleue.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/36px-Racine_carr%C3%A9e_bleue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/48px-Racine_carr%C3%A9e_bleue.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail:Mathématiques">Portail des mathématiques</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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