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href="/wiki/Portada" title="Visiteu la pàgina principal [z]" accesskey="z"><span>Portada</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Article_aleatori" title="Carrega una pàgina a l’atzar [x]" accesskey="x"><span>Article a l'atzar</span></a></li><li id="n-Articles-de-qualitat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Articles_de_qualitat"><span>Articles de qualitat</span></a></li> </ul> </div> </div> <div id="p-Comunitat" class="vector-menu mw-portlet mw-portlet-Comunitat" > <div class="vector-menu-heading"> Comunitat </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Portal" title="Sobre el projecte, què podeu fer, on trobareu les coses"><span>Portal viquipedista</span></a></li><li id="n-Agenda-d'actes" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Trobades"><span>Agenda d'actes</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:Canvis_recents" title="Una llista dels canvis recents al wiki [r]" accesskey="r"><span>Canvis recents</span></a></li><li id="n-La-taverna" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:La_taverna"><span>La taverna</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Contacte"><span>Contacte</span></a></li><li id="n-Xat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Canals_IRC"><span>Xat</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Ajuda" title="El lloc per a saber més coses"><span>Ajuda</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Portada" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Viquipèdia" 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class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Aparença" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Aparença</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ca.wikipedia.org&uselang=ca" class=""><span>Donatius</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Crea_compte&returnto=Ve%C3%AFnat+%28matem%C3%A0tiques%29" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori" class=""><span>Crea un compte</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Registre_i_entrada&returnto=Ve%C3%AFnat+%28matem%C3%A0tiques%29" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o" class=""><span>Inicia la sessió</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Més opcions" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Eines personals" > <label 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href="/w/index.php?title=Especial:Crea_compte&returnto=Ve%C3%AFnat+%28matem%C3%A0tiques%29" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crea un compte</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Registre_i_entrada&returnto=Ve%C3%AFnat+%28matem%C3%A0tiques%29" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Inicia la sessió</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"> Pàgines per a editors no registrats <a href="/wiki/Ajuda:Introducci%C3%B3" aria-label="Vegeu més informació sobre l'edició"><span>més informació</span></a> </div> <div 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[n]" accesskey="n"><span>Discussió per aquest IP</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="Lloc"> <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="Contingut" 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">Contingut</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">amaga</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">Inici</div> </a> </li> <li id="toc-Definició" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definició"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definició</span> </div> </a> <ul id="toc-Definició-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-En_un_espai_mètric" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#En_un_espai_mètric"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>En un espai mètric</span> </div> </a> <button aria-controls="toc-En_un_espai_mètric-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Commuta la subsecció En un espai mètric</span> </button> <ul id="toc-En_un_espai_mètric-sublist" class="vector-toc-list"> <li id="toc-Exemples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exemples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Exemples</span> </div> </a> <ul id="toc-Exemples-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definició_de_la_topologia_a_partir_dels_veïnats" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definició_de_la_topologia_a_partir_dels_veïnats"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Definició de la topologia a partir dels veïnats</span> </div> </a> <ul id="toc-Definició_de_la_topologia_a_partir_dels_veïnats-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Veïnat_perforat" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Veïnat_perforat"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Veïnat perforat</span> </div> </a> <ul id="toc-Veïnat_perforat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-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="Contingut" 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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Veïnat (matemàtiques)</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="Vés a un article en una altra llengua. Disponible en 40 llengües" > <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-40" 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">40 llengües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D9%88%D8%A7%D8%B1_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="جوار (رياضيات) - àrab" lang="ar" hreflang="ar" data-title="جوار (رياضيات)" data-language-autonym="العربية" data-language-local-name="àrab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%BA%D0%BE%D0%BB%D0%BD%D0%BE%D1%81%D1%82" title="Околност - búlgar" lang="bg" hreflang="bg" data-title="Околност" data-language-autonym="Български" data-language-local-name="búlgar" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Okolina_(matematika)" title="Okolina (matematika) - bosnià" lang="bs" hreflang="bs" data-title="Okolina (matematika)" data-language-autonym="Bosanski" data-language-local-name="bosnià" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%BE%D8%A7%D9%88%D8%B3%DB%8E%DB%8C%DB%8C_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="ھاوسێیی (ماتماتیک) - kurd central" lang="ckb" hreflang="ckb" data-title="ھاوسێیی (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="kurd central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Okol%C3%AD_(matematika)" title="Okolí (matematika) - txec" lang="cs" hreflang="cs" data-title="Okolí (matematika)" data-language-autonym="Čeština" data-language-local-name="txec" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D0%B0%D0%B2%D1%80%D0%B0%D0%BB%C4%83%D1%85_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Тавралăх (математика) - txuvaix" lang="cv" hreflang="cv" data-title="Тавралăх (математика)" data-language-autonym="Чӑвашла" data-language-local-name="txuvaix" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Umgebung_(Mathematik)" title="Umgebung (Mathematik) - alemany" lang="de" hreflang="de" data-title="Umgebung (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="alemany" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%B5%CE%B9%CF%84%CE%BF%CE%BD%CE%B9%CE%AC_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Γειτονιά (μαθηματικά) - grec" lang="el" hreflang="el" data-title="Γειτονιά (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics) - anglès" lang="en" hreflang="en" data-title="Neighbourhood (mathematics)" data-language-autonym="English" data-language-local-name="anglès" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/%C4%88irka%C5%ADa%C4%B5o" title="Ĉirkaŭaĵo - esperanto" lang="eo" hreflang="eo" data-title="Ĉirkaŭaĵo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Entorno_(matem%C3%A1tica)" title="Entorno (matemática) - espanyol" lang="es" hreflang="es" data-title="Entorno (matemática)" data-language-autonym="Español" data-language-local-name="espanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/%C3%9Cmbrus" title="Ümbrus - estonià" lang="et" hreflang="et" data-title="Ümbrus" data-language-autonym="Eesti" data-language-local-name="estonià" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D9%85%D8%B3%D8%A7%DB%8C%DA%AF%DB%8C_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="همسایگی (ریاضیات) - persa" lang="fa" hreflang="fa" data-title="همسایگی (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ymp%C3%A4rist%C3%B6_(topologia)" title="Ympäristö (topologia) - finès" lang="fi" hreflang="fi" data-title="Ympäristö (topologia)" data-language-autonym="Suomi" data-language-local-name="finès" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Voisinage_(math%C3%A9matiques)" title="Voisinage (mathématiques) - francès" lang="fr" hreflang="fr" data-title="Voisinage (mathématiques)" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Veci%C3%B1anza_(matem%C3%A1ticas)" title="Veciñanza (matemáticas) - gallec" lang="gl" hreflang="gl" data-title="Veciñanza (matemáticas)" data-language-autonym="Galego" data-language-local-name="gallec" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%91%D7%99%D7%91%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="סביבה (מתמטיקה) - hebreu" lang="he" hreflang="he" data-title="סביבה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="hebreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/K%C3%B6rnyezet_(matematika)" title="Környezet (matematika) - hongarès" lang="hu" hreflang="hu" data-title="Környezet (matematika)" data-language-autonym="Magyar" data-language-local-name="hongarès" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Lingkungan_(matematika)" title="Lingkungan (matematika) - indonesi" lang="id" hreflang="id" data-title="Lingkungan (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesi" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Grennd" title="Grennd - islandès" lang="is" hreflang="is" data-title="Grennd" data-language-autonym="Íslenska" data-language-local-name="islandès" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Intorno" title="Intorno - italià" lang="it" hreflang="it" data-title="Intorno" data-language-autonym="Italiano" data-language-local-name="italià" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%BF%91%E5%82%8D_(%E4%BD%8D%E7%9B%B8%E7%A9%BA%E9%96%93%E8%AB%96)" title="近傍 (位相空間論) - japonès" lang="ja" hreflang="ja" data-title="近傍 (位相空間論)" data-language-autonym="日本語" data-language-local-name="japonès" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9D%D2%AF%D0%BA%D1%82%D0%B5%D0%BD%D1%96%D2%A3_%D0%BC%D0%B0%D2%A3%D0%B0%D0%B9%D1%8B" title="Нүктенің маңайы - kazakh" lang="kk" hreflang="kk" data-title="Нүктенің маңайы" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B7%BC%EB%B0%A9" title="근방 - coreà" lang="ko" hreflang="ko" data-title="근방" data-language-autonym="한국어" data-language-local-name="coreà" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%B9%D0%BC%D0%B0%D0%BA_(%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Аймак (Математика) - kirguís" lang="ky" hreflang="ky" data-title="Аймак (Математика)" data-language-autonym="Кыргызча" data-language-local-name="kirguís" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Omgeving_(wiskunde)" title="Omgeving (wiskunde) - neerlandès" lang="nl" hreflang="nl" data-title="Omgeving (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandès" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Omegn_(matematikk)" title="Omegn (matematikk) - noruec bokmål" lang="nb" hreflang="nb" data-title="Omegn (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="noruec bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Otoczenie_i_s%C4%85siedztwo" title="Otoczenie i sąsiedztwo - polonès" lang="pl" hreflang="pl" data-title="Otoczenie i sąsiedztwo" data-language-autonym="Polski" data-language-local-name="polonès" 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/Vizinhan%C3%A7a_(matem%C3%A1tica)" title="Vizinhança (matemática) - portuguès" lang="pt" hreflang="pt" data-title="Vizinhança (matemática)" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Vecin%C4%83tate_(matematic%C4%83)" title="Vecinătate (matematică) - romanès" lang="ro" hreflang="ro" data-title="Vecinătate (matematică)" data-language-autonym="Română" data-language-local-name="romanès" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%BA%D1%80%D0%B5%D1%81%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Окрестность - rus" lang="ru" hreflang="ru" data-title="Окрестность" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Okolie_(matematika)" title="Okolie (matematika) - eslovac" lang="sk" hreflang="sk" data-title="Okolie (matematika)" data-language-autonym="Slovenčina" data-language-local-name="eslovac" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Okolica_(matematika)" title="Okolica (matematika) - eslovè" lang="sl" hreflang="sl" data-title="Okolica (matematika)" data-language-autonym="Slovenščina" data-language-local-name="eslovè" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D0%BA%D0%BE%D0%BB%D0%B8%D0%BD%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Околина (математика) - serbi" lang="sr" hreflang="sr" data-title="Околина (математика)" data-language-autonym="Српски / srpski" data-language-local-name="serbi" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Omgivning" title="Omgivning - suec" lang="sv" hreflang="sv" data-title="Omgivning" data-language-autonym="Svenska" data-language-local-name="suec" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kom%C5%9Fuluk_(matematik)" title="Komşuluk (matematik) - turc" lang="tr" hreflang="tr" data-title="Komşuluk (matematik)" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%BA%D1%96%D0%BB" title="Окіл - ucraïnès" lang="uk" hreflang="uk" data-title="Окіл" data-language-autonym="Українська" data-language-local-name="ucraïnès" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%A2n_c%E1%BA%ADn_(to%C3%A1n_h%E1%BB%8Dc)" title="Lân cận (toán học) - vietnamita" lang="vi" hreflang="vi" data-title="Lân cận (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%82%BB%E5%9F%9F" title="邻域 - xinès" lang="zh" hreflang="zh" data-title="邻域" data-language-autonym="中文" data-language-local-name="xinès" 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/%E9%84%B0%E5%9F%9F" title="鄰域 - xinès clàssic" lang="lzh" hreflang="lzh" data-title="鄰域" data-language-autonym="文言" data-language-local-name="xinès clàssic" 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/Q2478475#sitelinks-wikipedia" title="Modifica enllaços interlingües" class="wbc-editpage">Modifica els enllaços</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="Espais 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/Ve%C3%AFnat_(matem%C3%A0tiques)" title="Vegeu el contingut de la pàgina [c]" accesskey="c"><span>Pàgina</span></a></li><li id="ca-talk" 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href="/w/index.php?title=Especial:DownloadAsPdf&page=Ve%C3%AFnat_%28matem%C3%A0tiques%29&action=show-download-screen"><span>Baixa com a PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Ve%C3%AFnat_(matem%C3%A0tiques)&printable=yes" title="Versió per a impressió d'aquesta pàgina [p]" accesskey="p"><span>Versió per a impressora</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"> En altres projectes </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Neighborhood_(mathematics)" hreflang="en"><span>Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2478475" title="Enllaç a l'element del repositori de dades connectat [g]" accesskey="g"><span>Element a 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="Eines de la pàgina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparença"> <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">Aparença</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">amaga</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">De la Viquipèdia, l'enciclopèdia lliure</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="ca" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r34261971">.mw-parser-output .hatnote{width:100%;border-color:#77ccff;color:var(--color-base,#202122);background-color:#f5f5f5;margin-bottom:1em;font-style:italic}.mw-parser-output .hatnote i{font-style:normal}@media screen{html.skin-theme-clientpref-night .mw-parser-output .hatnote{color:var(--color-inverted,#fff);background-color:var(--background-color-inverted,#101418)}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .hatnote{color:var(--color-inverted,#fff);background-color:var(--background-color-inverted,#101418)}}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style> <table class="hatnote" cellspacing="5"> <tbody><tr> <td style="width: 25px; vertical-align: top;"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/22px-Disambig_grey.svg.png" decoding="async" width="22" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/33px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/44px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></span></span> </td> <td>Per a altres significats, vegeu «<a href="/wiki/Ve%C3%AFnat_(teoria_de_grafs)" title="Veïnat (teoria de grafs)">Veïnat (teoria de grafs)</a>». </td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Neighborhood_illust1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Neighborhood_illust1.svg/220px-Neighborhood_illust1.svg.png" decoding="async" width="220" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Neighborhood_illust1.svg/330px-Neighborhood_illust1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Neighborhood_illust1.svg/440px-Neighborhood_illust1.svg.png 2x" data-file-width="518" data-file-height="502" /></a><figcaption>Un conjunt <i>V</i> al pla és un veïnat d'un punt <i>p</i> si hi ha un <a href="/wiki/Obert_(matem%C3%A0tiques)" class="mw-redirect" title="Obert (matemàtiques)">obert</a> prou petit <i>B</i> que conté <i>p</i> i és contingut dins <i>V</i>.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Neighborhood_illust2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Neighborhood_illust2.svg/220px-Neighborhood_illust2.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Neighborhood_illust2.svg/330px-Neighborhood_illust2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Neighborhood_illust2.svg/440px-Neighborhood_illust2.svg.png 2x" data-file-width="480" data-file-height="360" /></a><figcaption>Un rectangle no és un veïnat de cap dels seus vèrtexs.</figcaption></figure> <p>En <a href="/wiki/Topologia" title="Topologia">topologia</a> i àrees relacionades de la <a href="/wiki/Matem%C3%A0tica" class="mw-redirect" title="Matemàtica">matemàtica</a>, un <b>veïnat</b> o <b>entorn</b> és un dels conceptes bàsics en un <a href="/wiki/Espai_topol%C3%B2gic" title="Espai topològic">espai topològic</a>. De manera intuïtiva, un veïnat d'un punt és un <a href="/wiki/Subconjunt" title="Subconjunt">subconjunt</a> que conté el punt i tots els punts prou propers al punt. Aquest concepte està estretament relacionat amb els conceptes d'<a href="/wiki/Obert_(matem%C3%A0tiques)" class="mw-redirect" title="Obert (matemàtiques)">obert</a> i <a href="/wiki/Interior_(topologia)" title="Interior (topologia)">interior</a> d'un conjunt. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definició"><span id="Definici.C3.B3"></span>Definició</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ve%C3%AFnat_(matem%C3%A0tiques)&action=edit&section=1" title="Modifica la secció: Definició"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si <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 \displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15912c0a3a4526142bcbc07f55d2c2d47e813d9d" 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 \displaystyle X}"></span> és un <a href="/wiki/Espai_topol%C3%B2gic" title="Espai topològic">espai topològic</a> i <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 \displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f187e02ee5adcf75f92b73790c8c2dfd333de8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \displaystyle p}"></span> és un punt 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 \displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15912c0a3a4526142bcbc07f55d2c2d47e813d9d" 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 \displaystyle X}"></span>, un <b>veïnat</b> 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 \displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f187e02ee5adcf75f92b73790c8c2dfd333de8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \displaystyle p}"></span> és un subconjunt <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 \displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eee5e9a28d2cbf3ecce810412bac885693a16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle \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 \displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15912c0a3a4526142bcbc07f55d2c2d47e813d9d" 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 \displaystyle X}"></span> que conté un subconjunt <a href="/wiki/Conjunt_obert" title="Conjunt obert">obert</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 \displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b09b353ddb3bd1ffad92a6a69e9507a1d45f80e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle \displaystyle U}"></span> que conté <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 \displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f187e02ee5adcf75f92b73790c8c2dfd333de8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \displaystyle p}"></span>: <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 \displaystyle p\in U\subset V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mi>U</mi> <mo>⊂</mo> <mi>V</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle p\in U\subset V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/688301340d81e918ad05ea58a3c69da09ac38545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.768ex; height:2.509ex;" alt="{\displaystyle \displaystyle p\in U\subset V}"></span>. Dit en altres termes, <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 \displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f187e02ee5adcf75f92b73790c8c2dfd333de8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \displaystyle p}"></span> és un punt interior 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 \displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eee5e9a28d2cbf3ecce810412bac885693a16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle \displaystyle V}"></span>. </p><p>Cal notar 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 \displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eee5e9a28d2cbf3ecce810412bac885693a16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle \displaystyle V}"></span> pot no ser obert. Quan <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 \displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eee5e9a28d2cbf3ecce810412bac885693a16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle \displaystyle V}"></span> és obert s'anomena <b>veïnat obert</b>. Cal parar atenció al fet que alguns autors requereixen en la definició de veïnat la condició de ser obert. </p><p>Un conjunt que és un veïnat de cadascun dels seus punts és obert, i recíprocament. </p><p>El conjunt de tots els veïnats d'un punt s'anomena <i>sistema de veïnats</i> del punt. Una <b>base de veïnats</b> d'un punt <i>p</i> és un conjunt de veïnats amb la propietat que qualsevol veïnat de <i>p</i> conté un veïnat de la base. </p><p>Si <i>S</i> és un <a href="/wiki/Subconjunt" title="Subconjunt">subconjunt</a> de <i>X</i>, un <b>veïnat</b> de <i>S</i> és un subconjunt <i>V</i> que conté un obert <i>U</i> que conté <i>S</i>. Això significa que <i>V</i> és un veïnat de cadascun dels punts de <i>S</i>. </p> <div class="mw-heading mw-heading2"><h2 id="En_un_espai_mètric"><span id="En_un_espai_m.C3.A8tric"></span>En un espai mètric</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ve%C3%AFnat_(matem%C3%A0tiques)&action=edit&section=2" title="Modifica la secció: En un espai mètric"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Neighborhood_illust3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Neighborhood_illust3.svg/220px-Neighborhood_illust3.svg.png" decoding="async" width="220" height="178" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Neighborhood_illust3.svg/330px-Neighborhood_illust3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Neighborhood_illust3.svg/440px-Neighborhood_illust3.svg.png 2x" data-file-width="518" data-file-height="419" /></a><figcaption>Un conjunt <i>S</i> al pla i un veïnat <i>V</i> de <i>S</i>.</figcaption></figure> <p>Sigui (<i>M</i>, <i>d</i>) un espai mètric, <i>p</i> un punt de <i>M</i>, i <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 B(p;r)=\{x\in M\mid d(p,x)<r\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>M</mi> <mo>∣</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(p;r)=\{x\in M\mid d(p,x)<r\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa0446d793100d1a01e61609b56e4ac36f65470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.504ex; height:2.843ex;" alt="{\displaystyle B(p;r)=\{x\in M\mid d(p,x)<r\}}"></span> la bola oberta de centre <i>p</i> i radi <i>r</i> (essent <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 r>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23cbbcd53bd13620bc53490e3eec42790850b452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r>0}"></span>). Aquestes boles obertes formen una base de veïnats de <i>p</i> en el sentit esmentat anteriorment: un conjunt <i>V</i> és un veïnat de <i>p</i> si hi ha una bola oberta <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 B(p;r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(p;r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a617623587d2980a66f39bff1606161510f892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.825ex; height:2.843ex;" alt="{\displaystyle B(p;r)}"></span> continguda en <i>V</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Exemples">Exemples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ve%C3%AFnat_(matem%C3%A0tiques)&action=edit&section=3" title="Modifica la secció: Exemples"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Considerem la <a href="/wiki/Nombre_real" title="Nombre real">recta real</a> <b>R</b> amb la seva distància usual. Els intervals tancat [-1,1] i obert (-1,1) són ambdós veïnats de l'origen. </p><p>Dins la <a href="/wiki/Nombre_real" title="Nombre real">recta real</a> <b>R</b>, considerem el conjunt <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=\bigcup _{n\in \mathbb {N} }B\left(n\,;\,1/2^{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mi>B</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\bigcup _{n\in \mathbb {N} }B\left(n\,;\,1/2^{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9487cf640c04ce3fa62e6df8042259017099724c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.411ex; height:5.676ex;" alt="{\displaystyle V=\bigcup _{n\in \mathbb {N} }B\left(n\,;\,1/2^{n}\right)}"></span>. Llavors <i>V</i> és un veïnat del conjunt <b>N</b> dels <a href="/wiki/Nombre_natural" title="Nombre natural">nombres naturals</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Definició_de_la_topologia_a_partir_dels_veïnats"><span id="Definici.C3.B3_de_la_topologia_a_partir_dels_ve.C3.AFnats"></span>Definició de la topologia a partir dels veïnats</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ve%C3%AFnat_(matem%C3%A0tiques)&action=edit&section=4" title="Modifica la secció: Definició de la topologia a partir dels veïnats"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La definició de veïnat donada anteriorment depèn del concepte previ de topologia. Tanmateix, es pot partir d'una definició abstracta de sistema de veïnats i definir a partir d'ella el concepte de conjunt obert. </p><p>Un sistema de veïnats en <i>X</i> és l'assignació d'un conjunt <i>N</i>(<i>x</i>) de parts de <i>X</i> a cada punt <i>x</i> de <i>X</i> de manera que </p> <ol><li>tota part de <i>X</i> que conté un dels conjunts de <i>N</i>(<i>x</i>) pertany a <i>N</i>(<i>x</i>)</li> <li>la intersecció de dos conjunts de <i>N</i>(<i>x</i>) pertany a <i>N</i>(<i>x</i>)</li> <li>el punt <i>x</i> pertany a cada conjunt <i>U</i> de <i>N</i>(<i>x</i>)</li> <li>cada <i>U</i> de <i>N</i>(<i>x</i>) conté un <i>V</i> de <i>N</i>(<i>x</i>) tal que, si <i>y</i> pertany a <i>V</i>, llavors <i>U</i> és de <i>N</i>(<i>y</i>).</li></ol> <p>Partint d'això, es defineix un conjunt obert en <i>X</i> com aquell que és un veïnat de cadascun dels seus punts. </p> <div class="mw-heading mw-heading2"><h2 id="Veïnat_perforat"><span id="Ve.C3.AFnat_perforat"></span>Veïnat perforat</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ve%C3%AFnat_(matem%C3%A0tiques)&action=edit&section=5" title="Modifica la secció: Veïnat perforat"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un <b>veïnat perforat</b> (o <i>veïnat reduït</i>) d'un punt <i>p</i> és un veïnat de <i>p</i> menys el propi <i>p</i>. Per exemple, en un espai mètric la <i>bola perforada</i> <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 B^{*}(p;r)=\{x\in M\mid 0<d(p,x)<r\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>;</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>M</mi> <mo>∣</mo> <mn>0</mn> <mo><</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B^{*}(p;r)=\{x\in M\mid 0<d(p,x)<r\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d81e2dc7eec33543d44ddb40fb1e9df0be52fe21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.819ex; height:2.843ex;" alt="{\displaystyle B^{*}(p;r)=\{x\in M\mid 0<d(p,x)<r\}}"></span> és un veïnat perforat de <i>p</i>. Evidentment, un tal conjunt <i>no</i> és un veïnat de <i>p</i>. El concepte de veïnat perforat és útil a l'hora de parlar de <a href="/wiki/L%C3%ADmit" title="Límit">límit</a> d'una funció. </p> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ve%C3%AFnat_(matem%C3%A0tiques)&action=edit&section=6" title="Modifica la secció: Bibliografia"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r33663753">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}.mw-parser-output .side-box-center{clear:both;margin:auto}}</style><div class="side-box metadata side-box-right plainlinks"> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">A <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/P%C3%A0gina_principal?uselang=ca">Wikimedia Commons</a></span> hi ha contingut multimèdia relatiu a: <i><b><a href="https://commons.wikimedia.org/wiki/Category:Neighborhood_(mathematics)" class="extiw" title="commons:Category:Neighborhood (mathematics)">Veïnat</a></b></i></div></div> </div> <ul><li><span class="citation book" style="font-style:normal" id="CITEREFKelley1975"><a href="/wiki/John_L._Kelley" title="John L. Kelley"><span style="font-variant: small-caps;">Kelley</span>, John L.</a> <i>General topology</i>.  New York: Springer-Verlag, 1975. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0387901256" title="Especial:Fonts bibliogràfiques/0387901256">ISBN 0387901256</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+topology&rft.aulast=Kelley&rft.aufirst=John+L.&rft.date=1975&rft.pub=New+York%3A+Springer-Verlag&rft.isbn=0387901256"><span style="display: none;"> </span></span></li> <li><span class="citation book" style="font-style:normal" id="CITEREFBredon1993"><span style="font-variant: small-caps;">Bredon</span>, Glen E. <i>Topology and geometry</i>.  New York: Springer-Verlag, 1993. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0387979263" title="Especial:Fonts bibliogràfiques/0387979263">ISBN 0387979263</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology+and+geometry&rft.aulast=Bredon&rft.aufirst=Glen+E.&rft.date=1993&rft.pub=New+York%3A+Springer-Verlag&rft.isbn=0387979263"><span style="display: none;"> </span></span></li> <li><span class="citation book" style="font-style:normal" id="CITEREFKaplansky2001"><a href="/wiki/Irving_Kaplansky" title="Irving Kaplansky"><span style="font-variant: small-caps;">Kaplansky</span>, Irving</a>. <i>Set Theory and Metric Spaces</i>.  American Mathematical Society, 2001. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0821826948" title="Especial:Fonts bibliogràfiques/0821826948">ISBN 0821826948</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory+and+Metric+Spaces&rft.aulast=Kaplansky&rft.aufirst=Irving&rft.date=2001&rft.pub=American+Mathematical+Society&rft.isbn=0821826948"><span style="display: none;"> </span></span></li> <li><span class="citation book" style="font-style:normal" id="CITEREFBourbaki1971"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki"><span style="font-variant: small-caps;">Bourbaki</span>, Nicolas</a>. <i>Topologie générale</i>.  Masson, 1971.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topologie+g%C3%A9n%C3%A9rale&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rft.date=1971&rft.pub=Masson"><span style="display: none;"> </span></span></li></ul> <p><span style="display: none;" class="interProject"><a href="https://ca.wiktionary.org/wiki/ve%C3%AFnat" class="extiw" title="wikt:veïnat">Viccionari</a></span> </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtingut de «<a dir="ltr" href="https://ca.wikipedia.org/w/index.php?title=Veïnat_(matemàtiques)&oldid=30857257">https://ca.wikipedia.org/w/index.php?title=Veïnat_(matemàtiques)&oldid=30857257</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categories</a>: <ul><li><a href="/wiki/Categoria:Topologia" title="Categoria:Topologia">Topologia</a></li><li><a href="/wiki/Categoria:An%C3%A0lisi_matem%C3%A0tica" title="Categoria:Anàlisi matemàtica">Anàlisi matemàtica</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoria oculta: <ul><li><a href="/wiki/Categoria:P%C3%A0gines_amb_enlla%C3%A7_commonscat_des_de_Wikidata" title="Categoria:Pàgines amb enllaç commonscat des de Wikidata">Pàgines amb enllaç commonscat des de Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La pàgina va ser modificada per darrera vegada el 30 oct 2022 a les 17:13.</li> <li id="footer-info-copyright">El text està disponible sota la <a href="/wiki/Viquip%C3%A8dia:Text_de_la_llic%C3%A8ncia_de_Creative_Commons_Reconeixement-Compartir_Igual_4.0_No_adaptada" title="Viquipèdia:Text de la llicència de Creative Commons Reconeixement-Compartir Igual 4.0 No adaptada"> Llicència de Creative Commons Reconeixement i Compartir-Igual</a>; es poden aplicar termes addicionals. 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