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class="vector-toc-link" href="#Proprietà"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Proprietà</span> </div> </a> <button aria-controls="toc-Proprietà-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>Attiva/disattiva la sottosezione Proprietà</span> </button> <ul id="toc-Proprietà-sublist" class="vector-toc-list"> <li id="toc-Struttura_topologica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Struttura_topologica"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Struttura topologica</span> </div> </a> <ul id="toc-Struttura_topologica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spazi_normati" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spazi_normati"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Spazi normati</span> </div> </a> <ul id="toc-Spazi_normati-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equivalenze" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equivalenze"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Equivalenze</span> </div> </a> <ul id="toc-Equivalenze-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distanza_tra_punti_e_insiemi_e_tra_insiemi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Distanza_tra_punti_e_insiemi_e_tra_insiemi"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Distanza tra punti e insiemi e tra insiemi</span> </div> </a> <ul id="toc-Distanza_tra_punti_e_insiemi_e_tra_insiemi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitatezza" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limitatezza"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Limitatezza</span> </div> </a> <ul id="toc-Limitatezza-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spazi_metrici_prodotto" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spazi_metrici_prodotto"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Spazi metrici prodotto</span> </div> </a> <ul id="toc-Spazi_metrici_prodotto-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Esempi_di_spazi_metrici" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Esempi_di_spazi_metrici"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Esempi di spazi metrici</span> </div> </a> <ul id="toc-Esempi_di_spazi_metrici-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Note</span> </div> </a> <ul id="toc-Note-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">9</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Spazio metrico</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="Vai a una voce in un'altra lingua. Disponibile in 61 lingue" > <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-61" 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">61 lingue</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/%D9%81%D8%B6%D8%A7%D8%A1_%D9%85%D8%AA%D8%B1%D9%8A" title="فضاء متري - arabo" lang="ar" hreflang="ar" data-title="فضاء متري" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Espaciu_m%C3%A9tricu" title="Espaciu métricu - asturiano" lang="ast" hreflang="ast" data-title="Espaciu métricu" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Метрично пространство - bulgaro" lang="bg" hreflang="bg" data-title="Метрично пространство" data-language-autonym="Български" data-language-local-name="bulgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_m%C3%A8tric" title="Espai mètric - catalano" lang="ca" hreflang="ca" data-title="Espai mètric" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DB%86%D8%B4%D8%A7%DB%8C%DB%8C%DB%8C_%D9%85%DB%95%D8%AA%D8%B1%DB%8C" title="بۆشاییی مەتری - curdo centrale" lang="ckb" hreflang="ckb" data-title="بۆشاییی مەتری" data-language-autonym="کوردی" data-language-local-name="curdo centrale" 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/Metrick%C3%BD_prostor" title="Metrický prostor - ceco" lang="cs" hreflang="cs" data-title="Metrický prostor" data-language-autonym="Čeština" data-language-local-name="ceco" 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%9C%D0%B5%D1%82%D1%80%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%C4%83_%D1%83%C3%A7%D0%BB%C4%83%D1%85" title="Метрикăллă уçлăх - ciuvascio" lang="cv" hreflang="cv" data-title="Метрикăллă уçлăх" data-language-autonym="Чӑвашла" data-language-local-name="ciuvascio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gofod_metrig" title="Gofod metrig - gallese" lang="cy" hreflang="cy" data-title="Gofod metrig" data-language-autonym="Cymraeg" data-language-local-name="gallese" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Metrisk_rum" title="Metrisk rum - danese" lang="da" hreflang="da" data-title="Metrisk rum" data-language-autonym="Dansk" data-language-local-name="danese" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Metrischer_Raum" title="Metrischer Raum - tedesco" lang="de" hreflang="de" data-title="Metrischer Raum" data-language-autonym="Deutsch" data-language-local-name="tedesco" 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%9C%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CF%8C%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Μετρικός χώρος - greco" lang="el" hreflang="el" data-title="Μετρικός χώρος" data-language-autonym="Ελληνικά" data-language-local-name="greco" 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/Metric_space" title="Metric space - inglese" lang="en" hreflang="en" data-title="Metric space" data-language-autonym="English" data-language-local-name="inglese" 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/Metrika_spaco" title="Metrika spaco - esperanto" lang="eo" hreflang="eo" data-title="Metrika spaco" 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/Espacio_m%C3%A9trico" title="Espacio métrico - spagnolo" lang="es" hreflang="es" data-title="Espacio métrico" data-language-autonym="Español" data-language-local-name="spagnolo" 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/Meetriline_ruum" title="Meetriline ruum - estone" lang="et" hreflang="et" data-title="Meetriline ruum" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Espazio_metriko" title="Espazio metriko - basco" lang="eu" hreflang="eu" data-title="Espazio metriko" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D9%85%D8%AA%D8%B1%DB%8C" title="فضای متری - persiano" lang="fa" hreflang="fa" data-title="فضای متری" data-language-autonym="فارسی" data-language-local-name="persiano" 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/Metrinen_avaruus" title="Metrinen avaruus - finlandese" lang="fi" hreflang="fi" data-title="Metrinen avaruus" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Meetriline_ruum" title="Meetriline ruum - võro" lang="vro" hreflang="vro" data-title="Meetriline ruum" data-language-autonym="Võro" data-language-local-name="võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_m%C3%A9trique" title="Espace métrique - francese" lang="fr" hreflang="fr" data-title="Espace métrique" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Sp%C3%A1s_m%C3%A9adrach" title="Spás méadrach - irlandese" lang="ga" hreflang="ga" data-title="Spás méadrach" data-language-autonym="Gaeilge" data-language-local-name="irlandese" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Espazo_m%C3%A9trico" title="Espazo métrico - galiziano" lang="gl" hreflang="gl" data-title="Espazo métrico" data-language-autonym="Galego" data-language-local-name="galiziano" 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%9E%D7%A8%D7%97%D7%91_%D7%9E%D7%98%D7%A8%D7%99" title="מרחב מטרי - ebraico" lang="he" hreflang="he" data-title="מרחב מטרי" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Metri%C4%8Dki_prostor" title="Metrički prostor - croato" lang="hr" hreflang="hr" data-title="Metrički prostor" data-language-autonym="Hrvatski" data-language-local-name="croato" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Metrikus_t%C3%A9r" title="Metrikus tér - ungherese" lang="hu" hreflang="hu" data-title="Metrikus tér" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A5%D5%BF%D6%80%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BF%D5%A1%D6%80%D5%A1%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Մետրիկական տարածություն - armeno" lang="hy" hreflang="hy" data-title="Մետրիկական տարածություն" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_metrik" title="Ruang metrik - indonesiano" lang="id" hreflang="id" data-title="Ruang metrik" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" 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/Fir%C3%B0r%C3%BAm" title="Firðrúm - islandese" lang="is" hreflang="is" data-title="Firðrúm" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%B7%9D%E9%9B%A2%E7%A9%BA%E9%96%93" title="距離空間 - giapponese" lang="ja" hreflang="ja" data-title="距離空間" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%94%E1%83%A2%E1%83%A0%E1%83%98%E1%83%99%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%A1%E1%83%98%E1%83%95%E1%83%A0%E1%83%AA%E1%83%94" title="მეტრიკული სივრცე - georgiano" lang="ka" hreflang="ka" data-title="მეტრიკული სივრცე" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%BA%D0%B5%D2%A3%D1%96%D1%81%D1%82%D1%96%D0%BA" title="Метрикалық кеңістік - kazako" lang="kk" hreflang="kk" data-title="Метрикалық кеңістік" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AE%E0%B3%86%E0%B2%9F%E0%B3%8D%E0%B2%B0%E0%B2%BF%E0%B2%95%E0%B3%8D_%E0%B2%86%E0%B2%95%E0%B2%BE%E0%B2%B6" title="ಮೆಟ್ರಿಕ್ ಆಕಾಶ - kannada" lang="kn" hreflang="kn" data-title="ಮೆಟ್ರಿಕ್ ಆಕಾಶ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간 - coreano" lang="ko" hreflang="ko" data-title="거리 공간" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Metresche_Raum" title="Metresche Raum - lussemburghese" lang="lb" hreflang="lb" data-title="Metresche Raum" data-language-autonym="Lëtzebuergesch" data-language-local-name="lussemburghese" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Metrin%C4%97_erdv%C4%97" title="Metrinė erdvė - lituano" lang="lt" hreflang="lt" data-title="Metrinė erdvė" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Метрички простор - macedone" lang="mk" hreflang="mk" data-title="Метрички простор" data-language-autonym="Македонски" data-language-local-name="macedone" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ruang_metrik" title="Ruang metrik - malese" lang="ms" hreflang="ms" data-title="Ruang metrik" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%90%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%B8%E1%80%86" title="အတိုင်းဆ - birmano" lang="my" hreflang="my" data-title="အတိုင်းဆ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmano" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Metrische_ruimte" title="Metrische ruimte - olandese" lang="nl" hreflang="nl" data-title="Metrische ruimte" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Metrisk_rom" title="Metrisk rom - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Metrisk rom" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Metrisk_rom" title="Metrisk rom - norvegese bokmål" lang="nb" hreflang="nb" data-title="Metrisk rom" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese 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/Przestrze%C5%84_metryczna" title="Przestrzeń metryczna - polacco" lang="pl" hreflang="pl" data-title="Przestrzeń metryczna" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Spassi_m%C3%A9trich" title="Spassi métrich - piemontese" lang="pms" hreflang="pms" data-title="Spassi métrich" data-language-autonym="Piemontèis" data-language-local-name="piemontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_m%C3%A9trico" title="Espaço métrico - portoghese" lang="pt" hreflang="pt" data-title="Espaço métrico" data-language-autonym="Português" data-language-local-name="portoghese" 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/Spa%C8%9Biu_metric" title="Spațiu metric - rumeno" lang="ro" hreflang="ro" data-title="Spațiu metric" data-language-autonym="Română" data-language-local-name="rumeno" 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%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Метрическое пространство - russo" lang="ru" hreflang="ru" data-title="Метрическое пространство" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Metric_space" title="Metric space - Simple English" lang="en-simple" hreflang="en-simple" data-title="Metric space" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Metrick%C3%BD_priestor" title="Metrický priestor - slovacco" lang="sk" hreflang="sk" data-title="Metrický priestor" data-language-autonym="Slovenčina" data-language-local-name="slovacco" 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/Metri%C4%8Dni_prostor" title="Metrični prostor - sloveno" lang="sl" hreflang="sl" data-title="Metrični prostor" data-language-autonym="Slovenščina" data-language-local-name="sloveno" 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%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Метрички простор - serbo" lang="sr" hreflang="sr" data-title="Метрички простор" data-language-autonym="Српски / srpski" data-language-local-name="serbo" 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/Metriskt_rum" title="Metriskt rum - svedese" lang="sv" hreflang="sv" data-title="Metriskt rum" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%86%E0%AE%9F%E0%AF%8D%E0%AE%B0%E0%AE%BF%E0%AE%95%E0%AF%8D_%E0%AE%B5%E0%AF%86%E0%AE%B3%E0%AE%BF" title="மெட்ரிக் வெளி - tamil" lang="ta" hreflang="ta" data-title="மெட்ரிக் வெளி" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Metrik_uzay" title="Metrik uzay - turco" lang="tr" hreflang="tr" data-title="Metrik uzay" data-language-autonym="Türkçe" data-language-local-name="turco" 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%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Метричний простір - ucraino" lang="uk" hreflang="uk" data-title="Метричний простір" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A8%D8%AD%D8%B1_%D9%81%D8%B6%D8%A7" title="بحر فضا - urdu" lang="ur" hreflang="ur" data-title="بحر فضا" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Spassio_m%C3%A8trico" title="Spassio mètrico - veneto" lang="vec" hreflang="vec" data-title="Spassio mètrico" data-language-autonym="Vèneto" data-language-local-name="veneto" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_m%C3%AAtric" title="Không gian mêtric - vietnamita" lang="vi" hreflang="vi" data-title="Không gian mêtric" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%97%B4" title="度量空间 - wu" lang="wuu" hreflang="wuu" data-title="度量空间" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%97%B4" title="度量空间 - cinese" lang="zh" hreflang="zh" data-title="度量空间" data-language-autonym="中文" data-language-local-name="cinese" 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%BA%A6%E9%87%8F%E7%A9%BA%E9%96%93" title="度量空間 - cinese classico" lang="lzh" hreflang="lzh" data-title="度量空間" data-language-autonym="文言" data-language-local-name="cinese classico" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%96%93" title="度量空間 - cantonese" lang="yue" hreflang="yue" data-title="度量空間" data-language-autonym="粵語" data-language-local-name="cantonese" class="interlanguage-link-target"><span>粵語</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/Q180953#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</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="Namespace"> <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/Spazio_metrico" title="Vedi la voce [c]" accesskey="c"><span>Voce</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussione:Spazio_metrico" rel="discussion" title="Vedi le discussioni relative a questa pagina [t]" accesskey="t"><span>Discussione</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" 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vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Spazio_metrico&action=edit" title="Modifica il wikitesto di questa pagina [e]" accesskey="e"><span>Modifica wikitesto</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Spazio_metrico&action=history"><span>Cronologia</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Generale </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speciale:PuntanoQui/Spazio_metrico" title="Elenco di tutte le pagine che sono collegate a questa [j]" accesskey="j"><span>Puntano qui</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speciale:ModificheCorrelate/Spazio_metrico" rel="nofollow" title="Elenco delle ultime modifiche alle pagine collegate a questa [k]" accesskey="k"><span>Modifiche correlate</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speciale:PagineSpeciali" title="Elenco di tutte le pagine speciali [q]" accesskey="q"><span>Pagine speciali</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Spazio_metrico&oldid=136768618" title="Collegamento permanente a questa versione di questa pagina"><span>Link permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Spazio_metrico&action=info" title="Ulteriori informazioni su questa pagina"><span>Informazioni pagina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speciale:Cita&page=Spazio_metrico&id=136768618&wpFormIdentifier=titleform" title="Informazioni su come citare questa pagina"><span>Cita questa voce</span></a></li><li id="t-urlshortener" class="mw-list-item"><a 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dell'archivio dati [g]" accesskey="g"><span>Elemento 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="Strumenti pagine"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspetto"> <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">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">nascondi</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">Da Wikipedia, l'enciclopedia libera.</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="it" dir="ltr"><p>Uno <b>spazio metrico</b> è un <a href="/wiki/Insieme" title="Insieme">insieme</a> di elementi, detti <i>punti</i>, nel quale è definita una <a href="/wiki/Distanza_(matematica)" title="Distanza (matematica)">distanza</a>, detta anche <i>metrica</i>. Lo spazio metrico più comune è lo <a href="/wiki/Spazio_euclideo" title="Spazio euclideo">spazio euclideo</a> di dimensione 1, 2 o 3. </p><p>Uno spazio metrico è in particolare uno <a href="/wiki/Spazio_topologico" title="Spazio topologico">spazio topologico</a>, e quindi eredita le nozioni di <a href="/wiki/Spazio_compatto" title="Spazio compatto">compattezza</a>, <a href="/wiki/Spazio_connesso" title="Spazio connesso">connessione</a>, <a href="/wiki/Insieme_aperto" title="Insieme aperto">insieme aperto</a> e <a href="/wiki/Insieme_chiuso" title="Insieme chiuso">chiuso</a>. Si applicano quindi agli spazi metrici gli strumenti della <a href="/wiki/Topologia_algebrica" title="Topologia algebrica">topologia algebrica</a>, quali ad esempio il <a href="/wiki/Gruppo_fondamentale" title="Gruppo fondamentale">gruppo fondamentale</a>. </p><p>Qualsiasi oggetto contenuto nello spazio euclideo è esso stesso uno spazio metrico. Molti insiemi di <a href="/wiki/Funzione_(matematica)" title="Funzione (matematica)">funzioni</a> sono dotati di una metrica: accade ad esempio se formano uno <a href="/wiki/Spazio_di_Hilbert" title="Spazio di Hilbert">spazio di Hilbert</a> o <a href="/wiki/Spazio_di_Banach" title="Spazio di Banach">di Banach</a>. Per questi motivi gli spazi metrici giocano un ruolo fondamentale in <a href="/wiki/Geometria" title="Geometria">geometria</a> e in <a href="/wiki/Analisi_funzionale" title="Analisi funzionale">analisi funzionale</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definizione">Definizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=1" title="Modifica la sezione Definizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=1" title="Edit section's source code: Definizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uno spazio metrico è una struttura <a href="/wiki/Matematica" title="Matematica">matematica</a> costituita da una coppia <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,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span> di elementi, dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> è un <a href="/wiki/Insieme" title="Insieme">insieme</a> e <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> una funzione <a href="/wiki/Distanza_(matematica)" title="Distanza (matematica)">distanza</a>, detta anche <i>metrica</i>, che associa a due punti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> e <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> un <a href="/wiki/Numero_reale" title="Numero reale">numero reale</a> non negativo <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 d(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3772957879a8bbf7946bddf5743c508a1d5072c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.544ex; height:2.843ex;" alt="{\displaystyle d(x,y)}"></span> in modo che le seguenti proprietà valgano per ogni scelta di <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,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>:<sup id="cite_ref-def_1-0" class="reference"><a href="#cite_note-def-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <ul><li><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 d(x,y)>0\iff x\neq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)>0\iff x\neq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66256d4829bcf28c552dbfa781ddb000cc5fc270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.286ex; height:2.843ex;" alt="{\displaystyle d(x,y)>0\iff x\neq y}"></span></li> <li><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 d(x,y)=0\iff x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=0\iff x=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32fd7ce57db3ea42e1bbb4aabb703cb709c6bfec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.286ex; height:2.843ex;" alt="{\displaystyle d(x,y)=0\iff x=y}"></span></li> <li><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 d(x,y)=d(y,x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=d(y,x)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6075ef47b0569ea999ac9abf8e97bbcd4da92bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.768ex; height:2.843ex;" alt="{\displaystyle d(x,y)=d(y,x)\ }"></span></li> <li><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 d(x,y)\leq d(x,z)+d(z,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)\leq d(x,z)+d(z,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5d6179f9dbf3ce34351feebb3698e28c973719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.263ex; height:2.843ex;" alt="{\displaystyle d(x,y)\leq d(x,z)+d(z,y)}"></span></li></ul> <p>L'ultima proprietà è detta <a href="/wiki/Disuguaglianza_triangolare" title="Disuguaglianza triangolare">disuguaglianza triangolare</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Proprietà"><span id="Propriet.C3.A0"></span>Proprietà</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=2" title="Modifica la sezione Proprietà" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=2" title="Edit section's source code: Proprietà"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Struttura_topologica">Struttura topologica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=3" title="Modifica la sezione Struttura topologica" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=3" title="Edit section's source code: Struttura topologica"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uno spazio metrico possiede naturalmente anche una struttura <a href="/wiki/Spazio_topologico" title="Spazio topologico">topologica</a>: l'insieme delle <a href="/wiki/Palla_(matematica)" title="Palla (matematica)">palle</a> aperte centrate nei vari punti avente raggio variabile fornisce infatti una sua <a href="/wiki/Base_(topologia)" title="Base (topologia)">base topologica</a>. </p><p>Esplicitamente, un insieme sarà <a href="/wiki/Insieme_aperto" title="Insieme aperto">aperto</a> se è l'<a href="/wiki/Unione_(insiemistica)" title="Unione (insiemistica)">unione</a> di un certo numero (finito o infinito) di palle. Uno spazio metrico è perciò, quasi per definizione, uno <a href="/wiki/Spazio_metrizzabile" title="Spazio metrizzabile">spazio metrizzabile</a>. </p><p>Per una <a href="/wiki/Funzione_(matematica)" title="Funzione (matematica)">funzione</a> definita in uno spazio metrico sarà possibile dunque parlare di <a href="/wiki/Funzione_continua" title="Funzione continua">continuità</a> e la definizione generale (usando le <a href="/wiki/Controimmagine" title="Controimmagine">controimmagini</a> degli aperti) potrà essere riformulata in funzione di dischi: </p> <dl><dd><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,d)\to (Y,d')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:(X,d)\to (Y,d')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eca2a7c9807e22bc30e3dad7278202f918a0bf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.388ex; height:3.009ex;" alt="{\displaystyle f:(X,d)\to (Y,d')}"></span> è continua in <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> se per ogni <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> esiste un <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 \delta (r)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (r)>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0431229dae0d56b2c205c905fd3b8be66b1610fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.168ex; height:2.843ex;" alt="{\displaystyle \delta (r)>0}"></span> tale che <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\in B_{d}(x_{0},\delta (r))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in B_{d}(x_{0},\delta (r))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c86d93dd740f11fcee6166426efd3de515aeaa85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.16ex; height:2.843ex;" alt="{\displaystyle x\in B_{d}(x_{0},\delta (r))}"></span> implica <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)\in B_{d'}(f(x_{0}),r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>d</mi> <mo>′</mo> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\in B_{d'}(f(x_{0}),r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdffe9a8b58f4f6be28fafef60d33f576da406eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.011ex; height:2.843ex;" alt="{\displaystyle f(x)\in B_{d'}(f(x_{0}),r)}"></span>,</dd></dl> <p>dove <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_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8152ba18cface59883b0d177b0295a249b6433db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.856ex; height:2.509ex;" alt="{\displaystyle B_{d}}"></span> (risp. <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_{d'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>d</mi> <mo>′</mo> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{d'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/529bc7d6b1fd4c8727f24cbc97c2ba89fdc5472d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.389ex; height:2.676ex;" alt="{\displaystyle B_{d'}}"></span>) rappresenta la palla nella metrica <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> (risp. <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 d'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f310a68106a9e308bdaf887ff8f7171c4cb9d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.509ex;" alt="{\displaystyle d'}"></span>). Scritta in un altro modo, questa definizione dice che: </p> <dl><dd><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,d)\to (Y,d')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:(X,d)\to (Y,d')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eca2a7c9807e22bc30e3dad7278202f918a0bf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.388ex; height:3.009ex;" alt="{\displaystyle f:(X,d)\to (Y,d')}"></span> è continua in <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> se per ogni <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> esiste un <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 \delta (r)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (r)>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0431229dae0d56b2c205c905fd3b8be66b1610fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.168ex; height:2.843ex;" alt="{\displaystyle \delta (r)>0}"></span> tale che <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 d(x,x_{0})<\delta (r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo><</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,x_{0})<\delta (r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/653cde64a7692507710e20b4596bc2878ac3db75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.778ex; height:2.843ex;" alt="{\displaystyle d(x,x_{0})<\delta (r)}"></span> implica <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 d'(f(x),f(x_{0}))<r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo><</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'(f(x),f(x_{0}))<r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f8db72294d3e0ef4488bc4ac8dc2ee92d478019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.783ex; height:3.009ex;" alt="{\displaystyle d'(f(x),f(x_{0}))<r}"></span>.</dd></dl> <p>Tale definizione è già molto vicina a quella usuale per funzioni <a href="/wiki/Numero_reale" title="Numero reale">reali</a>. </p><p>Addizionalmente, uno spazio metrico è anche uno <a href="/wiki/Spazio_uniforme" title="Spazio uniforme">spazio uniforme</a>, definendo un sottoinsieme <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"> <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/af0f6064540e84211d0ffe4dac72098adfa52845" 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 V}"></span> di <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 M\times M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>×</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\times M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/674e3ff5ea2de11c108d05ccd695a5ef97f1a215" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.725ex; height:2.176ex;" alt="{\displaystyle M\times M}"></span> essere un entourage se e solo se esiste un <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 \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span> tale che se <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 d(x,y)<\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)<\epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64e1e513bab784b06b54f6aa697342e41becd645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.587ex; height:2.843ex;" alt="{\displaystyle d(x,y)<\epsilon }"></span> allora <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,y)\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94a765d76f66c745840fbff14fe8fea1d10d51cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.956ex; height:2.843ex;" alt="{\displaystyle (x,y)\in V}"></span>. La struttura uniforme generalizza quella topologica. </p><p>È possibile costruire esempi semplici di metriche topologicamente equivalenti ma con strutture uniformi distinte: basta prendere, in <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></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 d_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cccb5a6a2f1acab4ca255e0be86c224ed82282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}"></span> la <a href="/wiki/Metrica_euclidea" class="mw-redirect" title="Metrica euclidea">metrica euclidea</a> e <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 d_{2}(x,y)=|e^{x}-e^{y}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}(x,y)=|e^{x}-e^{y}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a3e99dc2dae777a5220ad2c5a5840b96baebf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.213ex; height:2.843ex;" alt="{\displaystyle d_{2}(x,y)=|e^{x}-e^{y}|}"></span>; allora <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,y)\in \mathbb {R} ^{2}:|x-y|<1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(x,y)\in \mathbb {R} ^{2}:|x-y|<1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6aad5560cba5c07f8164b28ffaf2e643522bfec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.044ex; height:3.176ex;" alt="{\displaystyle \{(x,y)\in \mathbb {R} ^{2}:|x-y|<1\}}"></span> è un entourage nella struttura uniforme data da <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 d_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cccb5a6a2f1acab4ca255e0be86c224ed82282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}"></span> ma non in quella data da <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 d_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9276f8f68c5c23329de74ad76e69f6801358fb1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{2}}"></span>. Intuitivamente, la difformità è data dalla <i>distorsione</i> della metrica usuale secondo una funzione non <a href="/wiki/Continuit%C3%A0_uniforme" title="Continuità uniforme">uniformemente continua</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Spazi_normati">Spazi normati</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=4" title="Modifica la sezione Spazi normati" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=4" title="Edit section's source code: Spazi normati"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uno <a href="/wiki/Spazio_normato" title="Spazio normato">spazio vettoriale normato</a> </p> <dl><dd><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 (M,\|\cdot \|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,\|\cdot \|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1edc92ef37fc0ffde3003ca97f0232aad68ab816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.289ex; height:2.843ex;" alt="{\displaystyle (M,\|\cdot \|)}"></span></dd></dl> <p>è in modo naturale anche uno spazio metrico dotato della distanza </p> <dl><dd><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 d(x,y)=\|x-y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=\|x-y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94df456fedac6b08c33cb4dffa5345a0ce0891f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.293ex; height:2.843ex;" alt="{\displaystyle d(x,y)=\|x-y\|}"></span></dd></dl> <p>Le proprietà della distanza discendono infatti da quelle della <a href="/wiki/Norma_(matematica)" title="Norma (matematica)">norma</a>. </p><p>Uno <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazio vettoriale</a> munito di una <a href="/wiki/Seminorma" title="Seminorma">seminorma</a> genera invece una <a href="/wiki/Distanza_(matematica)#Generalizzazioni" title="Distanza (matematica)">pseudometrica</a>, cioè una funzione che può assegnare distanza nulla a punti diversi, e quindi <b>non</b> uno spazio metrico. Si può ovviare all'inconveniente introducendo la <a href="/wiki/Relazione_di_equivalenza" title="Relazione di equivalenza">relazione di equivalenza</a> ~, che identifica due punti se e solo se hanno distanza nulla. Passando dunque all'<a href="/wiki/Insieme_quoziente" class="mw-redirect" title="Insieme quoziente">insieme quoziente</a> </p> <dl><dd><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^{*}=X_{/\sim }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>∼</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{*}=X_{/\sim }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cab8b327b7602e0b99fb51820f2d789ce877834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.407ex; height:3.176ex;" alt="{\displaystyle X^{*}=X_{/\sim }}"></span></dd></dl> <p>e definendo, se <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> è la pseudometrica, </p> <dl><dd><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 d^{*}([x],[y])=d(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{*}([x],[y])=d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fda81c746e0e8fd5da138b85df63ce27d004b35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.831ex; height:2.843ex;" alt="{\displaystyle d^{*}([x],[y])=d(x,y)}"></span></dd></dl> <p>la funzione <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 d^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de21e957a2d59bc958c7b842fd35f2b5277728e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.272ex; height:2.343ex;" alt="{\displaystyle d^{*}}"></span> risulta essere, oltre che ben definita, proprio una metrica per <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"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </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/01924e6e5570e2631081fea6c6981b4872d3e04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{*}}"></span>. Il quoziente conserva la <a href="/wiki/Spazio_topologico" title="Spazio topologico">topologia</a> che la pseudometrica induce su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> (esattamente nello stesso modo in cui lo fa una metrica), cioè <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> è <a href="/wiki/Insieme_aperto" title="Insieme aperto">aperto</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> se e solo se <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 \pi (A)=[A]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>A</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (A)=[A]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16c06c4f361a7509e4003d7a538d4f3cb1e65469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.02ex; height:2.843ex;" alt="{\displaystyle \pi (A)=[A]}"></span> (ovvero i punti di A considerati a meno dell'equivalenza) è aperto in <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"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </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/01924e6e5570e2631081fea6c6981b4872d3e04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{*}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Equivalenze">Equivalenze</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=5" title="Modifica la sezione Equivalenze" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=5" title="Edit section's source code: Equivalenze"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una <a href="/wiki/Biiezione" class="mw-redirect" title="Biiezione">biiezione</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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> tra due spazi metrici <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 (M_{1},d_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M_{1},d_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94d3c0aca0772489db301d6c287dc52c5545bc75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.415ex; height:2.843ex;" alt="{\displaystyle (M_{1},d_{1})}"></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 (M_{2},d_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M_{2},d_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05a4c68a67ff10c60270286f08ee4917de29200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.415ex; height:2.843ex;" alt="{\displaystyle (M_{2},d_{2})}"></span> si dice </p> <ul><li>una <a href="/wiki/Isometria" title="Isometria">isometria</a> se <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 d_{2}(f(x),f(y))=d_{1}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}(f(x),f(y))=d_{1}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb738b19f7a02611c7a0e7c5d23018f8bfb8c94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.457ex; height:2.843ex;" alt="{\displaystyle d_{2}(f(x),f(y))=d_{1}(x,y)}"></span> per ogni <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,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></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 M_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577d686fc81d1d1eb3ae54e78aeee8957baf6718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{1}}"></span> e <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 M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5d4dffae5ee0db4cc433e252ee9ed7530e5cf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{2}}"></span> sono <i>isometrici</i>).</li> <li>una <a href="/wiki/Similitudine_(geometria)" title="Similitudine (geometria)">similitudine</a> se <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 d_{2}(f(x),f(y))=kd_{1}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}(f(x),f(y))=kd_{1}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f25c874c45752a0df09497543b08624d1b423d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.669ex; height:2.843ex;" alt="{\displaystyle d_{2}(f(x),f(y))=kd_{1}(x,y)}"></span> per qualche <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 k>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}"></span>, per ogni <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,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></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 M_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577d686fc81d1d1eb3ae54e78aeee8957baf6718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{1}}"></span> e <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 M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5d4dffae5ee0db4cc433e252ee9ed7530e5cf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{2}}"></span> sono <i>simili</i>).</li> <li>una <a href="/w/index.php?title=Uniformit%C3%A0&action=edit&redlink=1" class="new" title="Uniformità (la pagina non esiste)">uniformità</a> se è un <a href="/wiki/Isomorfismo" title="Isomorfismo">isomorfismo</a> tra <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 M_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577d686fc81d1d1eb3ae54e78aeee8957baf6718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{1}}"></span> e <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 M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5d4dffae5ee0db4cc433e252ee9ed7530e5cf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{2}}"></span> visti come <a href="/wiki/Spazio_uniforme" title="Spazio uniforme">spazi uniformi</a>.</li> <li>un <a href="/wiki/Omeomorfismo" title="Omeomorfismo">omeomorfismo</a> se è un <a href="/wiki/Isomorfismo" title="Isomorfismo">isomorfismo</a> tra <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 M_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577d686fc81d1d1eb3ae54e78aeee8957baf6718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{1}}"></span> e <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 M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5d4dffae5ee0db4cc433e252ee9ed7530e5cf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{2}}"></span> visti come spazi topologici (<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 M_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577d686fc81d1d1eb3ae54e78aeee8957baf6718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{1}}"></span> e <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 M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5d4dffae5ee0db4cc433e252ee9ed7530e5cf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{2}}"></span> sono <i>omeomorfi</i>).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Distanza_tra_punti_e_insiemi_e_tra_insiemi">Distanza tra punti e insiemi e tra insiemi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=6" title="Modifica la sezione Distanza tra punti e insiemi e tra insiemi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=6" title="Edit section's source code: Distanza tra punti e insiemi e tra insiemi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Oltre alla distanza tra punti, in uno spazio metrico si possono introdurre altri concetti accessori, come la distanza <i>tra un punto e un insieme</i>, definita come </p> <dl><dd><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 \delta (x,E)=\inf _{y\in E}d(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <mi>E</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x,E)=\inf _{y\in E}d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/537a1aa3f67a08111bd3a42519356eb70c08966f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.196ex; height:4.343ex;" alt="{\displaystyle \delta (x,E)=\inf _{y\in E}d(x,y)}"></span></dd></dl> <p>È <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 \delta (x,E)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x,E)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5060c2c969b941f692c5c3c7239cc4724832cd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.258ex; height:2.843ex;" alt="{\displaystyle \delta (x,E)=0}"></span> se e solo se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> appartiene alla <a href="/wiki/Chiusura_(topologia)" title="Chiusura (topologia)">chiusura</a> di <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>. Per questa funzione vale una versione generale della <a href="/wiki/Disuguaglianza_triangolare" title="Disuguaglianza triangolare">disuguaglianza triangolare</a>, cioè </p> <dl><dd><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 \delta (x,E)\leq d(x,y)+\delta (y,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x,E)\leq d(x,y)+\delta (y,E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/789890f889dad7081b2e28dc2bb3a3e671b5b5e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.303ex; height:2.843ex;" alt="{\displaystyle \delta (x,E)\leq d(x,y)+\delta (y,E)}"></span>.</dd></dl> <p>Si possono definire inoltre più <i>distanze tra insiemi</i>. </p> <ul><li>Una è definita come l'<a href="/wiki/Estremo_inferiore" class="mw-redirect" title="Estremo inferiore">estremo inferiore</a> della distanza tra due punti dei due insiemi:</li></ul> <dl><dd><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 d(E,F)=\inf _{x\in E,y\in F}d(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(E,F)=\inf _{x\in E,y\in F}d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe5b7b5e39add2497153e87087dac8663794af7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.499ex; height:4.343ex;" alt="{\displaystyle d(E,F)=\inf _{x\in E,y\in F}d(x,y)}"></span></dd></dl> <p>Questa definizione, che è molto intuitiva, si rivela però poco utile, perché è solo una <a href="/wiki/Distanza_(matematica)#Generalizzazioni" title="Distanza (matematica)">parametrica</a> simmetrica, cioè soddisfa solo la non negatività e l'"auto-distanza" nulla: due insiemi non coincidenti con <a href="/wiki/Intersezione_(insiemistica)" title="Intersezione (insiemistica)">intersezione</a> non vuota o che si toccano (cioè per esempio <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 [1,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f29a59e6206d7c2c77fcbb74d6295a233190475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.91ex; height:2.843ex;" alt="{\displaystyle [1,2)}"></span> e <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 (2,3]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2,3]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d314fd895c3ff8bbf663247da7475abf0f432307" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.91ex; height:2.843ex;" alt="{\displaystyle (2,3]}"></span>) hanno distanza nulla. </p> <ul><li>Una definizione migliore è stata data da <a href="/wiki/Felix_Hausdorff" title="Felix Hausdorff">Felix Hausdorff</a> ed è la seguente:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(A,B)=\max\{e(A,B),e(B,A)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi>e</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>e</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(A,B)=\max\{e(A,B),e(B,A)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a4b3ecb09ded2a7de4b9c005d16b1322e554f8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.065ex; height:2.843ex;" alt="{\displaystyle H(A,B)=\max\{e(A,B),e(B,A)\}}"></span>,</dd></dl> <p>dove per evitare notazioni pesanti si è indicato con <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="{\textstyle e(A,B)=\sup _{x\in A}\delta (x,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>e</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle e(A,B)=\sup _{x\in A}\delta (x,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc77b9c4e555ae38f679630741298739ef0d112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.908ex; height:2.843ex;" alt="{\textstyle e(A,B)=\sup _{x\in A}\delta (x,B)}"></span> l'<i>eccedenza</i> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> su <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> è detta proprio <i><a href="/wiki/Distanza_di_Hausdorff" title="Distanza di Hausdorff">distanza di Hausdorff</a></i> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> da <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. In generale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> è solo una pseudometrica: la sua restrizione ai sottoinsiemi <a href="/wiki/Insieme_chiuso" title="Insieme chiuso">chiusi</a> dello spazio metrico soddisfa però anche l'ultima proprietà mancante e la rende dunque una metrica su <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 P_{f}(X)=\{S\subseteq X:S\,{\mbox{chiuso}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> <mo>:</mo> <mi>S</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>chiuso</mtext> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{f}(X)=\{S\subseteq X:S\,{\mbox{chiuso}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628e74560016c515e52775fb37020d9275a9d26f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.585ex; height:3.009ex;" alt="{\displaystyle P_{f}(X)=\{S\subseteq X:S\,{\mbox{chiuso}}\}}"></span>, sottoclasse dell'<a href="/wiki/Insieme_delle_parti" title="Insieme delle parti">insieme delle parti</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Limitatezza">Limitatezza</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=7" title="Modifica la sezione Limitatezza" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=7" title="Edit section's source code: Limitatezza"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Insieme_limitato" title="Insieme limitato">Insieme limitato</a></b>.</span></div> </div> <p>Lo spazio metrico è la struttura più povera in cui si può cominciare a parlare di limitatezza di un insieme. Se <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 E\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4359fae89639fe08a113bd9154bff6b52047c7c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.854ex; height:2.343ex;" alt="{\displaystyle E\subseteq X}"></span>, allora <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> si dice limitato secondo la metrica presente <i>d</i> se esiste un raggio finito M tale che </p> <dl><dd><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 E\subset B_{d}(x,M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊂</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subset B_{d}(x,M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4aebe4485638eb1aeb2b434c0202b4f7b6bd337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.345ex; height:2.843ex;" alt="{\displaystyle E\subset B_{d}(x,M)}"></span> per qualche <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>.</dd></dl> <p>Ci sono altre definizioni equivalenti, cioè: </p> <ul><li>ponendo per definizione <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="{\textstyle diam(E)=\sup _{x,y\in E}d(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>E</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle diam(E)=\sup _{x,y\in E}d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f1fa17c2adf454ad7322c26c9ab9222b49adf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.203ex; height:3.176ex;" alt="{\textstyle diam(E)=\sup _{x,y\in E}d(x,y)}"></span> il <a href="/wiki/Diametro" title="Diametro">diametro</a> di <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>, se esso è un numero finito;</li> <li>se la sua <a href="/wiki/Chiusura_(topologia)" title="Chiusura (topologia)">chiusura</a> è limitata.</li></ul> <p>La nozione è però ovviamente dipendente dalla distanza che si pone sull'insieme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>: se per esempio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> è uno spazio illimitato con distanza <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>, esso ha diametro 1 nella distanza <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 d'={d \over 1+d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>d</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'={d \over 1+d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2eabbb34795f80161a7968856e30a393dd77a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.056ex; height:5.676ex;" alt="{\displaystyle d'={d \over 1+d}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Spazi_metrici_prodotto">Spazi metrici prodotto</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=8" title="Modifica la sezione Spazi metrici prodotto" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=8" title="Edit section's source code: Spazi metrici prodotto"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se <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},\dots ,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dots ,X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38ed92ce88f900210607bbb8f4d66e14d52d7a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\dots ,X_{n}}"></span> sono spazi metrici con distanze <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1},\dots ,g_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1},\dots ,g_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab736e4d81fa410e1a1f5971ce627bf1aa00a92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.669ex; height:2.009ex;" alt="{\displaystyle g_{1},\dots ,g_{n}}"></span> rispettivamente allora si può definire una metrica nel <a href="/wiki/Prodotto_cartesiano" title="Prodotto cartesiano">prodotto cartesiano</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 X_{1}\times \dots \times X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}\times \dots \times X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff15ea6ed20dbacd07a87b7e2994db36f9a3282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.525ex; height:2.509ex;" alt="{\displaystyle X_{1}\times \dots \times X_{n}}"></span> tra <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 {\vec {x}}=(x_{1},\dots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}=(x_{1},\dots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a328cea1d32a254b4491c73047468ea335b1c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.348ex; height:2.843ex;" alt="{\displaystyle {\vec {x}}=(x_{1},\dots ,x_{n})}"></span> e <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 {\vec {y}}=(y_{1},\dots ,y_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {y}}=(y_{1},\dots ,y_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c57d2104f58638cfaa662cefeba31cd795db5cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.902ex; height:2.843ex;" alt="{\displaystyle {\vec {y}}=(y_{1},\dots ,y_{n})}"></span> come </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g_{1}\times \dots \times g_{n})({\vec {x}},{\vec {y}}):=\sum _{i=1}^{n}{1 \over 2^{i}}{g_{i}(x_{i},y_{i}) \over 1+g_{i}(x_{i},y_{i})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g_{1}\times \dots \times g_{n})({\vec {x}},{\vec {y}}):=\sum _{i=1}^{n}{1 \over 2^{i}}{g_{i}(x_{i},y_{i}) \over 1+g_{i}(x_{i},y_{i})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11912623b032e8f26f4474bee96035b303c0c358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.086ex; height:6.843ex;" alt="{\displaystyle (g_{1}\times \dots \times g_{n})({\vec {x}},{\vec {y}}):=\sum _{i=1}^{n}{1 \over 2^{i}}{g_{i}(x_{i},y_{i}) \over 1+g_{i}(x_{i},y_{i})}}"></span>.</dd></dl> <p>La formula può essere estesa anche per prodotti <a href="/wiki/Insieme_numerabile" title="Insieme numerabile">numerabili</a>. </p><p>In generale, se <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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> è una <a href="/wiki/Norma_(matematica)" title="Norma (matematica)">norma</a> in <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} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>, allora si può definire la <i>metrica normata</i> nel prodotto cartesiano come </p> <dl><dd><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 N(d_{1},\dots ,d_{n}){\Big (}(x_{1},\dots ,x_{n}),(y_{1},\ldots ,y_{n}){\Big )}=N{\Big (}d_{1}(x_{1},y_{1}),\ldots ,d_{n}(x_{n},y_{n}){\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(d_{1},\dots ,d_{n}){\Big (}(x_{1},\dots ,x_{n}),(y_{1},\ldots ,y_{n}){\Big )}=N{\Big (}d_{1}(x_{1},y_{1}),\ldots ,d_{n}(x_{n},y_{n}){\Big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60305e87cbd9ecff874c485eb4551e3b812e8ae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:73.985ex; height:4.843ex;" alt="{\displaystyle N(d_{1},\dots ,d_{n}){\Big (}(x_{1},\dots ,x_{n}),(y_{1},\ldots ,y_{n}){\Big )}=N{\Big (}d_{1}(x_{1},y_{1}),\ldots ,d_{n}(x_{n},y_{n}){\Big )}}"></span></dd></dl> <p>e la topologia generata è coerente con la <a href="/wiki/Topologia_prodotto" title="Topologia prodotto">topologia prodotto</a>. </p><p>Come caso particolare, se <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 n=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}"></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 X_{1}=X_{2}=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}=X_{2}=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64b68d46539eefa70480bb415b45fe10de032b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.134ex; height:2.509ex;" alt="{\displaystyle X_{1}=X_{2}=X}"></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 d_{1}=d_{2}=d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}=d_{2}=d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/564b0f4a4b8103f4c77b5e54bb5ac8e0dfdfc079" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.939ex; height:2.509ex;" alt="{\displaystyle d_{1}=d_{2}=d}"></span> allora viene fuori che la funzione distanza <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 d\colon X\times X\to \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\colon X\times X\to \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43a60abe0925792a644fa0cfefba22b98372e60a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.853ex; height:2.509ex;" alt="{\displaystyle d\colon X\times X\to \mathbb {R} ^{+}}"></span> è <a href="/wiki/Continuit%C3%A0_uniforme" title="Continuità uniforme">uniformemente continua</a> rispetto ogni metrica normata e dunque è una <a href="/wiki/Funzione_continua" title="Funzione continua">funzione continua</a> rispetto alla topologia prodotto su <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\times X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0008a48abb9837a4cb0f495dd85d0ffda22ead92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.8ex; height:2.176ex;" alt="{\displaystyle X\times X}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Esempi_di_spazi_metrici">Esempi di spazi metrici</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=9" title="Modifica la sezione Esempi di spazi metrici" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=9" title="Edit section's source code: Esempi di spazi metrici"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Lo <a href="/wiki/Spazio_euclideo" title="Spazio euclideo">spazio euclideo</a> con la <a href="/wiki/Distanza_euclidea" title="Distanza euclidea">normale nozione di distanza</a>.</li> <li>Un insieme qualsiasi con la distanza definita nel modo seguente: la distanza tra due punti è <a href="/wiki/Uno" class="mw-redirect" title="Uno">1</a> se i punti sono diversi, <a href="/wiki/Zero" class="mw-redirect" title="Zero">0</a> altrimenti; in questo caso si dice distanza discreta.</li> <li>L'insieme delle <a href="/wiki/Funzione_continua" title="Funzione continua">funzioni continue</a> nell'<a href="/wiki/Intervallo_(matematica)" title="Intervallo (matematica)">intervallo</a> [0,1] è metrizzabile con la seguente metrica: date due funzioni <i>f</i><sub>1</sub>, <i>f</i><sub>2</sub> della variabile <i>x</i> il numero <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 d=\max |f_{1}(x)-f_{2}(x)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=\max |f_{1}(x)-f_{2}(x)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5582c933ffed23f3467352c9930f1d5a8da7fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.826ex; height:2.843ex;" alt="{\displaystyle d=\max |f_{1}(x)-f_{2}(x)|}"></span> è la distanza tra esse.</li> <li>Un sottoinsieme di uno spazio metrico si può considerare anch'esso in modo naturale uno spazio metrico: basta munirlo della opportuna restrizione della funzione distanza dello spazio di partenza. Quindi qualsiasi sottoinsieme dello spazio euclideo è un esempio di spazio metrico.</li> <li>Ogni <a href="/wiki/Spazio_normato" title="Spazio normato">spazio normato</a> è uno spazio metrico, dove la distanza tra due punti <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,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></span> è data dalla norma del vettore <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-y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3129cb3620bd9f38d0304a0fca719644d7d2d265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle x-y}"></span>. In questi casi si dice che la metrica è indotta dalla norma. Non vale però il viceversa, esistono cioè spazi metrici la cui metrica non può derivare da una norma, come mostra il prossimo esempio.</li> <li>L'<a href="/wiki/Numero_reale" title="Numero reale">insieme dei numeri reali</a>, con la distanza data da</li></ul> <dl><dd><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 d(x,y)=|\arctan(x)-\arctan(y)|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=|\arctan(x)-\arctan(y)|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d111e54fda783e134f5c91e2455b990477d7ddea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.847ex; height:2.843ex;" alt="{\displaystyle d(x,y)=|\arctan(x)-\arctan(y)|.}"></span></dd></dl> <p>Questa distanza, diversa da quella standard, non può essere indotta da una norma, in quanto non è invariante per <a href="/wiki/Traslazione_(geometria)" title="Traslazione (geometria)">traslazioni</a> (ovvero <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 d(x+z,y+z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x+z,y+z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c891f4df54337f757d017f62377a7160b49e49c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.401ex; height:2.843ex;" alt="{\displaystyle d(x+z,y+z)}"></span> è in generale diversa da <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 d(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3772957879a8bbf7946bddf5743c508a1d5072c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.544ex; height:2.843ex;" alt="{\displaystyle d(x,y)}"></span>), mentre tutte le distanze indotte da norme lo sono. </p> <ul><li>Se <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,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span> è uno spazio metrico, allora è possibile definire una nuova metrica <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 d_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cccb5a6a2f1acab4ca255e0be86c224ed82282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}"></span> su <i>X</i> tale che qualunque coppia di punti di <i>X</i> abbia distanza minore o uguale a 1. Basta infatti prendere</li></ul> <dl><dd><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 d_{1}(x,y)={\frac {d(x,y)}{d(x,y)+1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}(x,y)={\frac {d(x,y)}{d(x,y)+1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07c0dfc834ddc12692f9a97f8c0bda931f8b936d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.72ex; height:6.509ex;" alt="{\displaystyle d_{1}(x,y)={\frac {d(x,y)}{d(x,y)+1}}.}"></span></dd></dl> <p>Si può verificare che <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 d_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cccb5a6a2f1acab4ca255e0be86c224ed82282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}"></span> è ancora una metrica su <i>X</i>. Inoltre se <i>X</i> è <a href="/wiki/Insieme_limitato" title="Insieme limitato">illimitato</a> rispetto alla metrica <i>d</i>, risulta avere <a href="/wiki/Diametro" title="Diametro">diametro</a> 1 nella metrica <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 d_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cccb5a6a2f1acab4ca255e0be86c224ed82282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}"></span>, ovvero risulta <a href="/wiki/Insieme_limitato" title="Insieme limitato">limitato</a> nella metrica <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 d_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cccb5a6a2f1acab4ca255e0be86c224ed82282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}"></span>. La nozione di limitatezza di un insieme non è dunque un concetto "assoluto". </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=10" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=10" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-def-1"><a href="#cite_ref-def_1-0"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFrudin">W. Rudin</a>, Pag. 9</cite>.</span> </li> </ol></div> <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=Spazio_metrico&veaction=edit&section=11" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=11" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Athanase Papadopoulos, <i>Metric Spaces, Convexity and Nonpositive Curvature</i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">European Mathematical Society</a>, 2004, SBN 978-3-03719-010-4.</li> <li><cite id="CITEREFrudin" class="citation libro" style="font-style:normal"> Walter Rudin, <span style="font-style:italic;">Real and Complex Analysis</span>, Mladinska Knjiga, McGraw-Hill, 1970, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-07-054234-1" title="Speciale:RicercaISBN/0-07-054234-1">0-07-054234-1</a>.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=12" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=12" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Spazio_pseudometrico" title="Spazio pseudometrico">Spazio pseudometrico</a></li> <li><a href="/wiki/Spazio_ultrametrico" title="Spazio ultrametrico">Spazio ultrametrico</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=13" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=13" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a href="https://it.wikiversity.org/wiki/Spazi_metrici" class="extiw" title="v:Spazi metrici">Wikiversità</a></li> <li class="" title=""><span class="plainlinks" title="commons:Category:Metric geometry"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Metric_geometry?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://it.wikiversity.org/wiki/" title="Collabora a Wikiversità"><img alt="Collabora a Wikiversità" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/18px-Wikiversity_logo_2017.svg.png" decoding="async" width="18" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/36px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span> <a href="https://it.wikiversity.org/wiki/" class="extiw" title="v:">Wikiversità</a> contiene risorse su <b><a href="https://it.wikiversity.org/wiki/Spazi_metrici" class="extiw" title="v:Spazi metrici">spazio metrico</a></b></li> <li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file su <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Metric_geometry?uselang=it">spazio metrico</a></span></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spazio_metrico&veaction=edit&section=14" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spazio_metrico&action=edit&section=14" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFEnciclopedia_della_scienza_e_della_tecnica" class="citation libro" style="font-style:normal"> Luca Tomassini, <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/spazio-metrico_(Enciclopedia-della-Scienza-e-della-Tecnica)/"><span style="font-style:italic;">spazio metrico</span></a>, in <span style="font-style:italic;">Enciclopedia della scienza e della tecnica</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2007-2008.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q180953#P10037" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFEnciclopedia_della_Matematica" class="citation libro" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/spazio-metrico_(Enciclopedia-della-Matematica)/"><span style="font-style:italic;">spazio metrico</span></a>, in <span style="font-style:italic;">Enciclopedia della Matematica</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2013.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q180953#P9621" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFBritannica.com" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Stephan C. Carlson, <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/metric-space"><span style="font-style:italic;">metric space</span></a>, su <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a></span>, Encyclopædia Britannica, Inc.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q180953#P1417" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFOpen_Library" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://openlibrary.org/subjects/metric_spaces"><span style="font-style:italic;">Opere riguardanti Metric spaces</span></a>, su <span style="font-style:italic;"><a href="/wiki/Open_Library" class="mw-redirect" title="Open Library">Open Library</a></span>, <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q180953#P3847" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. 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font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Metric_space"><span style="font-style:italic;">Metric space</span></a>, su <span style="font-style:italic;"><a href="/wiki/Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a></span>, Springer e European Mathematical Society.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q180953#P7554" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFFOLDOC" class="citation testo" style="font-style:normal">(<span style="font-weight:bolder; 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padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Topologia" title="Template:Topologia"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Topologia&action=edit&redlink=1" class="new" title="Discussioni template:Topologia (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Topologia&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a href="/wiki/Topologia" title="Topologia">Topologia</a></span></th></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;">Concetti di <a href="/wiki/Topologia_generale" title="Topologia generale">Topologia generale</a></th><td colspan="1" class="navbox_list navbox_odd"><table class="subnavbox"><tbody><tr><td colspan="2" class="navbox_center"><a href="/wiki/Spazio_topologico" title="Spazio topologico">Spazio topologico</a><b> ·</b> <a href="/wiki/Base_(topologia)" title="Base (topologia)">Base</a><b> ·</b> <a href="/wiki/Prebase" title="Prebase">Prebase</a><b> ·</b> <a href="/wiki/Ricoprimento" title="Ricoprimento">Ricoprimento</a><b> ·</b> <a href="/wiki/Assiomi_di_chiusura_di_Kuratowski" title="Assiomi di chiusura di Kuratowski">Assiomi di chiusura di Kuratowski</a><b> ·</b> <a href="/wiki/Invariante_topologico" title="Invariante topologico">Invariante topologico</a><b> ·</b> <a href="/wiki/Relazione_di_finezza" title="Relazione di finezza">Relazione di finezza</a><b> ·</b> <a href="/wiki/Partizione_dell%27unit%C3%A0" title="Partizione dell'unità">Partizione dell'unità</a><b> ·</b> <a href="/wiki/Propriet%C3%A0_dell%27intersezione_finita" title="Proprietà dell'intersezione finita">Proprietà dell'intersezione finita</a></td></tr><tr><th class="subnavbox_group">Sottoinsiemi</th><td colspan="1"><a href="/wiki/Intervallo_(matematica)" title="Intervallo (matematica)">Intervallo</a><b> ·</b> <a href="/wiki/Insieme_aperto" title="Insieme aperto">Aperto</a><b> ·</b> <a href="/wiki/Intorno" title="Intorno">Intorno</a><b> ·</b> <a href="/wiki/Insieme_chiuso" title="Insieme chiuso">Chiuso</a><b> ·</b> <a href="/wiki/Insieme_localmente_chiuso" title="Insieme localmente chiuso">Insieme localmente chiuso</a><b> ·</b> <a href="/wiki/Insieme_chiuso-aperto" title="Insieme chiuso-aperto">Insieme chiuso-aperto</a><b> ·</b> <a href="/wiki/Parte_interna" title="Parte interna">Parte interna</a><b> ·</b> <a href="/wiki/Chiusura_(topologia)" title="Chiusura (topologia)">Chiusura</a><b> ·</b> <a href="/wiki/Frontiera_(topologia)" title="Frontiera (topologia)">Frontiera</a><b> ·</b> <a href="/wiki/Insieme_derivato" title="Insieme derivato">Insieme derivato</a><b> ·</b> <a href="/wiki/Insieme_limite" title="Insieme limite">Insieme limite</a><b> ·</b> <a href="/wiki/Insieme_perfetto" title="Insieme perfetto">Insieme perfetto</a><b> ·</b> <a href="/wiki/Insieme_denso" title="Insieme denso">Insieme denso</a><b> ·</b> <a href="/wiki/Insieme_mai_denso" title="Insieme mai denso">Insieme mai denso</a></td></tr><tr><th class="subnavbox_group">Punti</th><td colspan="1"><a href="/wiki/Punto_isolato" title="Punto isolato">Punto isolato</a><b> ·</b> <a href="/wiki/Punto_di_accumulazione" title="Punto di accumulazione">Punto di accumulazione</a><b> ·</b> <a href="/wiki/Punto_di_aderenza" title="Punto di aderenza">Punto di aderenza</a></td></tr><tr><th class="subnavbox_group">Funzioni</th><td colspan="1"><a href="/wiki/Funzione_continua" title="Funzione continua">Funzione continua</a><b> ·</b> <a href="/wiki/Omeomorfismo" title="Omeomorfismo">Omeomorfismo</a><b> ·</b> <a href="/wiki/Funzione_aperta" title="Funzione aperta">Funzione aperta</a><b> ·</b> <a href="/wiki/Funzione_chiusa" title="Funzione chiusa">Funzione chiusa</a><b> ·</b> <a href="/wiki/Funzione_propria" title="Funzione propria">Funzione propria</a><b> ·</b> <a href="/wiki/Contrazione_(matematica)" title="Contrazione (matematica)">Contrazione</a><b> ·</b> <a href="/wiki/Retrazione" title="Retrazione">Retrazione</a><b> ·</b> <a href="/wiki/Germe_di_funzione" title="Germe di funzione">Germe di funzione</a><b> ·</b> <a href="/wiki/Funzione_a_supporto_compatto" title="Funzione a supporto compatto">Funzione a supporto compatto</a></td></tr><tr><th class="subnavbox_group">Successioni</th><td colspan="1"><a href="/wiki/Limite_(matematica)" title="Limite (matematica)">Limite</a><b> ·</b> <a href="/wiki/Limite_di_una_successione" title="Limite di una successione">Limite di una successione</a><b> ·</b> <a href="/wiki/Successione_(matematica)" title="Successione (matematica)">Successione</a><b> ·</b> <a href="/wiki/Rete_(matematica)" title="Rete (matematica)">Rete</a><b> ·</b> <a href="/wiki/Convergenza" title="Convergenza">Convergenza</a><b> ·</b> <a href="/wiki/Successione_di_Cauchy" title="Successione di Cauchy">Successione di Cauchy</a></td></tr><tr><th class="subnavbox_group">Teoremi</th><td colspan="1"><a href="/wiki/Teorema_di_Weierstrass" title="Teorema di Weierstrass">Teorema di Weierstrass</a><b> ·</b> <a href="/wiki/Teorema_di_Heine-Borel" title="Teorema di Heine-Borel">Heine-Borel</a><b> ·</b> <a href="/wiki/Teorema_di_Tichonov" title="Teorema di Tichonov">Tichonov</a><b> ·</b> <a href="/wiki/Lemma_del_tubo" title="Lemma del tubo">Lemma del tubo</a><b> ·</b> <a href="/wiki/Lemma_di_Urysohn" title="Lemma di Urysohn">Urysohn</a><b> ·</b> <a href="/wiki/Teorema_di_estensione_di_Tietze" title="Teorema di estensione di Tietze">Tietze</a><b> ·</b> <a href="/wiki/Teorema_della_categoria_di_Baire" title="Teorema della categoria di Baire">Baire</a><b> ·</b> <a href="/wiki/Teorema_del_punto_fisso_di_Brouwer" title="Teorema del punto fisso di Brouwer">Brouwer</a><b> ·</b> <a href="/wiki/Teoremi_di_punto_fisso" title="Teoremi di punto fisso">punto fisso</a><b> ·</b> <a href="/wiki/Teorema_di_Borsuk" title="Teorema di Borsuk">Teorema di Borsuk</a><b> ·</b> <a href="/wiki/Teorema_di_Borsuk-Ulam" title="Teorema di Borsuk-Ulam">Teorema di Borsuk-Ulam</a><b> ·</b> <a href="/wiki/Teorema_della_curva_di_Jordan" title="Teorema della curva di Jordan">Teorema della curva di Jordan</a><b> ·</b> <a href="/wiki/Teorema_della_mappa_di_Riemann" title="Teorema della mappa di Riemann">Teorema della mappa di Riemann</a></td></tr><tr><th class="subnavbox_group">Applicazioni pratiche</th><td colspan="1"><a href="/wiki/Topologia_dello_spazio-tempo" title="Topologia dello spazio-tempo">Topologia dello spazio-tempo</a><b> ·</b> <a href="/wiki/Teoria_quantistica_dei_campi_topologica" title="Teoria quantistica dei campi topologica">Teoria quantistica dei campi topologica</a><b> ·</b> <a href="/wiki/K-teoria_ritorta" title="K-teoria ritorta">K-teoria ritorta</a><b> ·</b> <a href="/wiki/Topologia_di_rete" title="Topologia di rete">Topologia di rete</a><b> ·</b> <a href="/wiki/Controllo_della_topologia" title="Controllo della topologia">Controllo della topologia</a><b> ·</b> <a href="/wiki/Topologia_molecolare" title="Topologia molecolare">Topologia molecolare</a></td></tr></tbody></table></td><td rowspan="6" class="navbox_image"><span typeof="mw:File"><a href="/wiki/File:Torus.jpg" class="mw-file-description" title="Toro"><img alt="Toro" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Torus.jpg/100px-Torus.jpg" decoding="async" width="100" height="56" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Torus.jpg/150px-Torus.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Torus.jpg/200px-Torus.jpg 2x" data-file-width="300" data-file-height="168" /></a></span></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;">Spazi topologici</th><td colspan="1" class="navbox_list navbox_even"><table class="subnavbox"><tbody><tr><th class="subnavbox_group">Topologie classiche</th><td colspan="1"><a href="/wiki/Topologia_banale" title="Topologia banale">Topologia banale</a><b> ·</b> <a href="/wiki/Spazio_di_Sierpi%C5%84ski" title="Spazio di Sierpiński">Spazio di Sierpiński</a><b> ·</b> <a href="/wiki/Topologia_cofinita" title="Topologia cofinita">Cofinita</a><b> ·</b> <a href="/wiki/Topologia_della_semicontinuit%C3%A0_inferiore" title="Topologia della semicontinuità inferiore">Topologia della semicontinuità inferiore</a><b> ·</b> <a href="/wiki/Topologia_di_Zariski" title="Topologia di Zariski">di Zariski</a><b> ·</b> <a href="/wiki/Spazio_euclideo#Topologia_euclidea" title="Spazio euclideo">Euclidea</a><b> ·</b> <a href="/wiki/Topologia_del_limite_inferiore" title="Topologia del limite inferiore">del limite inferiore</a> o di Sorgenfrey<b> ·</b> <a href="/wiki/Topologia_discreta" title="Topologia discreta">Discreta</a><b> ·</b> <a href="/wiki/Topologia_degli_interi_equispaziati" title="Topologia degli interi equispaziati">Topologia degli interi equispaziati</a><b> ·</b> <a href="/wiki/Numero_reale#Insieme_reale_esteso" title="Numero reale">Insieme reale esteso</a><b> ·</b> <a href="/wiki/Ordine_totale#Topologia_di_ordine" title="Ordine totale">Topologia di ordine</a><b> ·</b> <a href="/wiki/Piano_di_Moore" title="Piano di Moore">Piano di Moore</a><b> ·</b> <a href="/wiki/Numero_p-adico" title="Numero p-adico">Topologia p-adica</a></td></tr><tr><th class="subnavbox_group">Costruzioni topologiche</th><td colspan="1"><a href="/wiki/Topologia_prodotto" title="Topologia prodotto">Topologia prodotto</a><b> ·</b> <a href="/wiki/Topologia_di_sottospazio" title="Topologia di sottospazio">Topologia di sottospazio</a><b> ·</b> <a href="/wiki/Topologia_quoziente" title="Topologia quoziente">Topologia quoziente</a><b> ·</b> <a href="/wiki/Compattificazione" title="Compattificazione">Compattificazione</a> (<a href="/wiki/Compattificazione_di_Alexandrov" title="Compattificazione di Alexandrov">di Alexandrov</a><b> ·</b> <a href="/wiki/Compattificazione_di_Stone-%C4%8Cech" title="Compattificazione di Stone-Čech">di Stone-Čech</a>)<b> ·</b> <a href="/wiki/Cono_(topologia)" title="Cono (topologia)">Cono</a><b> ·</b> <a href="/wiki/Bouquet_(topologia)" title="Bouquet (topologia)">Bouquet</a><b> ·</b> <a href="/wiki/Rosa_(topologia)" title="Rosa (topologia)">Rosa</a><b> ·</b> <a href="/wiki/Sospensione_(matematica)" title="Sospensione (matematica)">Sospensione</a></td></tr><tr><th class="subnavbox_group">Topologie in <a href="/wiki/Analisi_funzionale" title="Analisi funzionale">Analisi funzionale</a></th><td colspan="1"><a href="/wiki/Spazio_funzionale" title="Spazio funzionale">Spazio funzionale</a><b> ·</b> <a href="/wiki/Topologia_iniziale" title="Topologia iniziale">Topologia iniziale</a> o debole<b> ·</b> <a href="/wiki/Topologia_operatoriale" title="Topologia operatoriale">Topologia operatoriale</a><b> ·</b> <a href="/wiki/Topologia_finale" title="Topologia finale">Topologia finale</a> o forte<b> ·</b> <a href="/wiki/Topologia_di_Mackey" title="Topologia di Mackey">Topologia di Mackey</a><b> ·</b> <a href="/wiki/Topologia_polare" title="Topologia polare">Topologia polare</a><b> ·</b> <a href="/wiki/Topologie_operatoriali_debole_e_forte" title="Topologie operatoriali debole e forte">Topologie operatoriali debole e forte</a></td></tr><tr><th class="subnavbox_group">Altri oggetti topologici</th><td colspan="1"><a href="/wiki/Sfera" title="Sfera">Sfera</a><b> ·</b> <a href="/wiki/Palla_(matematica)" title="Palla (matematica)">Palla</a><b> ·</b> <a href="/wiki/Toro_(geometria)" title="Toro (geometria)">Toro</a><b> ·</b> <a href="/wiki/Corpo_con_manici" title="Corpo con manici">Corpo con manici</a><b> ·</b> <a href="/wiki/Bottiglia_di_Klein" title="Bottiglia di Klein">Bottiglia di Klein</a><b> ·</b> <a href="/wiki/Bottiglia_di_Klein_solida" title="Bottiglia di Klein solida">Bottiglia di Klein solida</a><b> ·</b> <a href="/wiki/Anello_(topologia)" title="Anello (topologia)">Anello</a><b> ·</b> <a href="/wiki/Nastro_di_M%C3%B6bius" title="Nastro di Möbius">Nastro di Möbius</a><b> ·</b> <a href="/wiki/Retta_proiettiva" title="Retta proiettiva">Retta proiettiva</a><b> ·</b> <a href="/wiki/Piano_proiettivo" title="Piano proiettivo">Piano proiettivo</a><b> ·</b> <a href="/wiki/Superficie_di_Riemann" title="Superficie di Riemann">Superficie di Riemann</a><b> ·</b> <a href="/wiki/Teoria_dei_nodi" title="Teoria dei nodi">Nodo</a><b> ·</b> <a href="/wiki/Nodo_torico" title="Nodo torico">Nodo torico</a><b> ·</b> <a href="/wiki/Link_(teoria_dei_nodi)" title="Link (teoria dei nodi)">Link</a> <table class="subnavbox"><tbody><tr><th class="subnavbox_group"><a href="/wiki/Frattale" title="Frattale">Frattali</a></th><td colspan="1"><a href="/wiki/Insieme_di_Cantor" title="Insieme di Cantor">Insieme di Cantor</a><b> ·</b> <a href="/wiki/Spazio_di_Cantor" title="Spazio di Cantor">Spazio di Cantor</a><b> ·</b> <a href="/wiki/Polvere_di_Cantor" title="Polvere di Cantor">Polvere di Cantor</a><b> ·</b> <a href="/wiki/Spugna_di_Menger" title="Spugna di Menger">Spugna di Menger</a><b> ·</b> <a href="/wiki/Sfera_di_Alexander" title="Sfera di Alexander">Sfera di Alexander</a><b> ·</b> <a href="/wiki/Curva_di_Peano" title="Curva di Peano">Curva di Peano</a><b> ·</b> <a href="/wiki/Laghi_di_Wada" title="Laghi di Wada">Laghi di Wada</a></td></tr></tbody></table></td></tr><tr><th class="subnavbox_group">Strutture miste</th><td colspan="1"><a href="/wiki/Spazio_vettoriale_topologico" title="Spazio vettoriale topologico">Spazio vettoriale topologico</a><b> ·</b> <a href="/wiki/Gruppo_topologico" title="Gruppo topologico">Gruppo topologico</a><b> ·</b> <a href="/wiki/Gruppo_di_Lie" title="Gruppo di Lie">Gruppo di Lie</a><b> ·</b> <a href="/wiki/Spazio_uniforme" title="Spazio uniforme">Spazio uniforme</a><b> ·</b> <a href="/wiki/Algebra_di_Borel" title="Algebra di Borel">Algebra di Borel</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;"><a href="/wiki/Propriet%C3%A0_(matematica)" title="Proprietà (matematica)">Proprietà</a> degli spazi topologici</th><td colspan="1" class="navbox_list navbox_odd"><table class="subnavbox"><tbody><tr><th class="subnavbox_group">Numerabilità</th><td colspan="1"><a href="/wiki/Assioma_di_numerabilit%C3%A0" title="Assioma di numerabilità">Assioma di numerabilità</a><b> ·</b> <a href="/wiki/Spazio_primo-numerabile" title="Spazio primo-numerabile">Spazio primo-numerabile</a><b> ·</b> <a href="/wiki/Spazio_separabile" title="Spazio separabile">Spazio separabile</a><b> ·</b> <a href="/wiki/Spazio_sequenziale" title="Spazio sequenziale">Spazio sequenziale</a></td></tr><tr><th class="subnavbox_group">Separazione</th><td colspan="1"><a href="/wiki/Assioma_di_separazione" title="Assioma di separazione">Assioma di separazione</a><b> ·</b> <a href="/wiki/Spazio_T0" title="Spazio T0">Spazio T0</a><b> ·</b> <a href="/wiki/Spazio_T1" title="Spazio T1">Spazio T1</a><b> ·</b> <a href="/wiki/Spazio_di_Hausdorff" title="Spazio di Hausdorff">Spazio di Hausdorff</a><b> ·</b> <a href="/wiki/Spazio_regolare" title="Spazio regolare">Spazio regolare</a><b> ·</b> <a href="/wiki/Spazio_di_Tichonov" title="Spazio di Tichonov">Spazio di Tichonov</a><b> ·</b> <a href="/wiki/Spazio_normale" title="Spazio normale">Spazio normale</a></td></tr><tr><th class="subnavbox_group">Compattezza</th><td colspan="1"><a href="/wiki/Spazio_compatto" title="Spazio compatto">Spazio compatto</a><b> ·</b> <a href="/wiki/Spazio_paracompatto" title="Spazio paracompatto">Spazio paracompatto</a><b> ·</b> <a href="/wiki/Spazio_localmente_compatto" title="Spazio localmente compatto">Spazio localmente compatto</a><b> ·</b> <a href="/wiki/Spazio_di_Lindel%C3%B6f" title="Spazio di Lindelöf">Spazio di Lindelöf</a><b> ·</b> <a href="/wiki/Sottospazio_relativamente_compatto" title="Sottospazio relativamente compatto">Sottospazio relativamente compatto</a><b> ·</b> <a href="/wiki/Immersione_compatta" title="Immersione compatta">Immersione compatta</a></td></tr><tr><th class="subnavbox_group">Connessione</th><td colspan="1"><a href="/wiki/Spazio_connesso" title="Spazio connesso">Spazio connesso</a><b> ·</b> <a href="/wiki/Spazio_semplicemente_connesso" title="Spazio semplicemente connesso">Spazio semplicemente connesso</a></td></tr><tr><th class="subnavbox_group">Metrizzabilità</th><td colspan="1"><a class="mw-selflink selflink">Spazio metrico</a><b> ·</b> <a href="/wiki/Spazio_metrico_completo" title="Spazio metrico completo">Spazio metrico completo</a><b> ·</b> <a href="/wiki/Spazio_metrizzabile" title="Spazio metrizzabile">Spazio metrizzabile</a><b> ·</b> <a href="/wiki/Spazio_ultrametrico" title="Spazio ultrametrico">Spazio ultrametrico</a><b> ·</b> <a href="/wiki/Spazio_pseudometrico" title="Spazio pseudometrico">Spazio pseudometrico</a><b> ·</b> <a href="/wiki/Spazio_polacco" title="Spazio polacco">Spazio polacco</a><b> ·</b> <a href="/wiki/Spazio_normato" title="Spazio normato">Spazio normato</a><b> ·</b> <a href="/wiki/Spazio_totalmente_limitato" title="Spazio totalmente limitato">Spazio totalmente limitato</a></td></tr><tr><th class="subnavbox_group">Altre proprietà</th><td colspan="1"><a href="/wiki/Spazio_di_Baire" title="Spazio di Baire">Spazio di Baire</a><b> ·</b> <a href="/wiki/Spazio_topologico_noetheriano" title="Spazio topologico noetheriano">Spazio topologico noetheriano</a><b> ·</b> <a href="/wiki/Spazio_omogeneo" title="Spazio omogeneo">Spazio omogeneo</a><b> ·</b> <a href="/wiki/Orientazione" title="Orientazione">Orientazione</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;"><a href="/wiki/Topologia_differenziale" title="Topologia differenziale">Topologia differenziale</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Variet%C3%A0_(geometria)" title="Varietà (geometria)">Varietà</a> (<a href="/wiki/Variet%C3%A0_differenziabile" title="Varietà differenziabile">differenziabile</a><b> ·</b> <a href="/wiki/Variet%C3%A0_parallelizzabile" title="Varietà parallelizzabile">parallelizzabile</a><b> ·</b> <a href="/wiki/3-variet%C3%A0" title="3-varietà">3-varietà</a><b> ·</b> <a href="/wiki/3-variet%C3%A0_irriducibile" title="3-varietà irriducibile">3-varietà irriducibile</a>)<b> ·</b> <a href="/wiki/Atlante_(topologia)" title="Atlante (topologia)">Atlante</a><b> ·</b> <a href="/wiki/Diffeomorfismo" title="Diffeomorfismo">Diffeomorfismo</a> (<a href="/wiki/Diffeomorfismo_locale" title="Diffeomorfismo locale">locale</a><b> ·</b> <a href="/wiki/Diffeomorfismo_di_Anosov" title="Diffeomorfismo di Anosov">di Anosov</a>)<b> ·</b> <a href="/wiki/Immersione_(geometria)" title="Immersione (geometria)">Immersione</a><b> ·</b> <a href="/wiki/Curva_(matematica)" title="Curva (matematica)">Curva</a><b> ·</b> <a href="/wiki/Superficie" title="Superficie">Superficie</a><b> ·</b> <a href="/wiki/Campo_vettoriale" title="Campo vettoriale">Campo vettoriale</a><b> ·</b> <a href="/wiki/Fibrato" title="Fibrato">Fibrato</a> (<a href="/wiki/Fibrato_principale" title="Fibrato principale">principale</a><b> ·</b> <a href="/wiki/Fibrato_vettoriale" title="Fibrato vettoriale">vettoriale</a><b> ·</b> <a href="/wiki/Variet%C3%A0_fibrata" title="Varietà fibrata">Varietà fibrata</a>)<b> ·</b> <a href="/wiki/Fibrato_tangente" title="Fibrato tangente">Fibrato tangente</a><b> ·</b> <a href="/wiki/Spazio_tangente" title="Spazio tangente">Spazio tangente</a><b> ·</b> <a href="/wiki/Fibrazione_di_Hopf" title="Fibrazione di Hopf">Fibrazione di Hopf</a><b> ·</b> <a href="/wiki/Variet%C3%A0_con_bordo" title="Varietà con bordo">Varietà con bordo</a><b> ·</b> <a href="/wiki/Teorema_dell%27intorno_tubolare" title="Teorema dell'intorno tubolare">Teorema dell'intorno tubolare</a><b> ·</b> <a href="/wiki/Somma_connessa" title="Somma connessa">Somma connessa</a><b> ·</b> <a href="/wiki/Teorema_di_Kneser-Milnor" title="Teorema di Kneser-Milnor">Teorema di Kneser-Milnor</a><b> ·</b> <a href="/wiki/Congettura_di_geometrizzazione_di_Thurston" title="Congettura di geometrizzazione di Thurston">Congettura di geometrizzazione di Thurston</a><b> ·</b> <a href="/wiki/Cobordismo" title="Cobordismo">Cobordismo</a><b> ·</b> <a href="/wiki/Dimensione_topologica" title="Dimensione topologica">Dimensione topologica</a><b> ·</b> <a href="/wiki/Topologia_in_dimensione_bassa" title="Topologia in dimensione bassa">Topologia in dimensione bassa</a><b> ·</b> <a href="/wiki/Chirurgia_di_Dehn" title="Chirurgia di Dehn">Chirurgia di Dehn</a><b> ·</b> <a href="/wiki/Trasversalit%C3%A0" title="Trasversalità">Trasversalità</a><b> ·</b> <a href="/wiki/Eversione_della_sfera" title="Eversione della sfera">Eversione della sfera</a><b> ·</b> <a href="/wiki/Teoria_delle_foliazioni" title="Teoria delle foliazioni">Teoria delle foliazioni</a><b> ·</b> <a href="/wiki/Decomposizione_JSJ" title="Decomposizione JSJ">Decomposizione JSJ</a></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;"><a href="/wiki/Topologia_algebrica" title="Topologia algebrica">Topologia algebrica</a></th><td colspan="1" class="navbox_list navbox_odd"><table class="subnavbox"><tbody><tr><th class="subnavbox_group">Fondamenti</th><td colspan="1"><a href="/wiki/Spazio_semplicemente_connesso" title="Spazio semplicemente connesso">Spazio semplicemente connesso</a><b> ·</b> <a href="/wiki/Gruppo_fondamentale" title="Gruppo fondamentale">Gruppo fondamentale</a></td></tr><tr><th class="subnavbox_group">Omotopia</th><td colspan="1"><a href="/wiki/Arco_(topologia)" title="Arco (topologia)">Arco</a><b> ·</b> <a href="/wiki/Nerbo_(matematica)" title="Nerbo (matematica)">Nerbo</a><b> ·</b> <a href="/wiki/Omotopia" title="Omotopia">Omotopia</a><b> ·</b> <a href="/wiki/Gruppi_di_omotopia" title="Gruppi di omotopia">Gruppi di omotopia</a></td></tr><tr><th class="subnavbox_group">Omologia e coomologia</th><td colspan="1"><a href="/wiki/Omologia_(topologia)" title="Omologia (topologia)">Omologia</a><b> ·</b> <a href="/wiki/Omologia_singolare" title="Omologia singolare">Omologia singolare</a><b> ·</b> <a href="/wiki/Omologia_ciclica" title="Omologia ciclica">Omologia ciclica</a><b> ·</b> <a href="/wiki/Algebra_omologica" title="Algebra omologica">Algebra omologica</a><b> ·</b> <a href="/wiki/Coomologia_di_De_Rham" title="Coomologia di De Rham">Coomologia di De Rham</a><b> ·</b> <a href="/wiki/Categoria_abeliana" title="Categoria abeliana">Categoria abeliana</a></td></tr><tr><th class="subnavbox_group">Sollevamento</th><td colspan="1"><a href="/wiki/Sollevamento_(matematica)" title="Sollevamento (matematica)">Sollevamento</a><b> ·</b> <a href="/wiki/Teorema_del_sollevamento_dell%27omotopia" title="Teorema del sollevamento dell'omotopia">Teorema del sollevamento dell'omotopia</a><b> ·</b> <a href="/wiki/Teorema_di_unicit%C3%A0_del_sollevamento" title="Teorema di unicità del sollevamento">Teorema di unicità del sollevamento</a><b> ·</b> <a href="/wiki/Teorema_di_Van_Kampen" title="Teorema di Van Kampen">Teorema di Van Kampen</a></td></tr><tr><th class="subnavbox_group">Topologia algebrica avanzata</th><td colspan="1"><a href="/wiki/Grado_topologico" title="Grado topologico">Grado topologico</a><b> ·</b> <a href="/wiki/Indice_di_avvolgimento" title="Indice di avvolgimento">Indice di avvolgimento</a><b> ·</b> <a href="/wiki/Indice_di_un_campo_vettoriale" title="Indice di un campo vettoriale">Indice di un campo vettoriale</a><b> ·</b> <a href="/wiki/Rivestimento_(topologia)" title="Rivestimento (topologia)">Rivestimento</a><b> ·</b> <a href="/wiki/Numero_di_Betti" title="Numero di Betti">Numero di Betti</a><b> ·</b> <a href="/wiki/Successione_di_Mayer-Vietoris" title="Successione di Mayer-Vietoris">Successione di Mayer-Vietoris</a><b> ·</b> <a href="/wiki/Successione_esatta" title="Successione esatta">Successione esatta</a><b> ·</b> <a href="/wiki/Successione_spettrale" title="Successione spettrale">Successione spettrale</a><b> ·</b> <a href="/wiki/Complesso_simpliciale" title="Complesso simpliciale">Complesso simpliciale</a><b> ·</b> <a href="/wiki/Complesso_di_celle" title="Complesso di celle">Complesso di celle</a><b> ·</b> <a href="/wiki/Complesso_di_catene" title="Complesso di catene">Complesso di catene</a><b> ·</b> <a href="/wiki/Schema_simpliciale" title="Schema simpliciale">Schema simpliciale</a></td></tr><tr><th class="subnavbox_group">Superfici</th><td colspan="1"><a href="/wiki/Caratteristica_di_Eulero" title="Caratteristica di Eulero">Caratteristica di Eulero</a><b> ·</b> <a href="/wiki/Formula_di_Eulero_per_i_poliedri" title="Formula di Eulero per i poliedri">Formula di Eulero per i poliedri</a><b> ·</b> <a href="/wiki/Genere_(matematica)" title="Genere (matematica)">Genere</a><b> ·</b> <a href="/wiki/Taglio_(topologia)" title="Taglio (topologia)">Taglio</a><b> ·</b> <a href="/wiki/Superficie_incompressibile" title="Superficie incompressibile">Superficie incompressibile</a><b> ·</b> <a href="/wiki/Classificazione_delle_superfici" title="Classificazione delle superfici">Classificazione delle superfici</a><b> ·</b> <a href="/wiki/Mapping_class_group" title="Mapping class group">Mapping class group</a><b> ·</b> <a href="/wiki/Teorema_della_palla_pelosa" title="Teorema della palla pelosa">Teorema della palla pelosa</a><b> ·</b> <a href="/wiki/Teorema_di_Poincar%C3%A9-Hopf" title="Teorema di Poincaré-Hopf">Teorema di Poincaré-Hopf</a><b> ·</b> <a href="/wiki/Congettura_di_Poincar%C3%A9" title="Congettura di Poincaré">Congettura di Poincaré</a><b> ·</b> <a href="/wiki/Congettura_di_Hodge" title="Congettura di Hodge">Congettura di Hodge</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:right;">Topologi di rilievo</th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a><b> ·</b> <a href="/wiki/Felix_Hausdorff" title="Felix Hausdorff">Felix Hausdorff</a><b> ·</b> <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a><b> ·</b> <a href="/wiki/Eduard_%C4%8Cech" title="Eduard Čech">Eduard Čech</a><b> ·</b> <a href="/wiki/John_Milnor" title="John Milnor">John Milnor</a><b> ·</b> <a href="/w/index.php?title=Pierre_Samuel&action=edit&redlink=1" class="new" title="Pierre Samuel (la pagina non esiste)">Pierre Samuel</a><b> ·</b> <a href="/w/index.php?title=Norman_Steenrod&action=edit&redlink=1" class="new" title="Norman Steenrod (la pagina non esiste)">Norman Steenrod</a><b> ·</b> <a href="/wiki/Ren%C3%A9_Thom" title="René Thom">René Thom</a><b> ·</b> <a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a><b> ·</b> <a href="/wiki/Andrej_Nikolaevi%C4%8D_Kolmogorov" title="Andrej Nikolaevič Kolmogorov">Andrej Nikolaevič Kolmogorov</a><b> ·</b> <a href="/wiki/Stephen_Smale" title="Stephen Smale">Stephen Smale</a><b> ·</b> <a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Michael Atiyah</a><b> ·</b> <a href="/wiki/William_Thurston" title="William Thurston">William Thurston</a><b> ·</b> <a href="/wiki/Marston_Morse" title="Marston Morse">Marston Morse</a><b> ·</b> <a href="/wiki/Luitzen_Brouwer" title="Luitzen Brouwer">Luitzen Brouwer</a></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .CdA th{background-color:#202122}}</style><table class="CdA"><tbody><tr><th><a href="/wiki/Aiuto:Controllo_di_autorit%C3%A0" title="Aiuto:Controllo di autorità">Controllo di autorità</a></th><td><a href="/wiki/Nuovo_soggettario" title="Nuovo soggettario">Thesaurus BNCF</a> <span class="uid"><a rel="nofollow" class="external text" href="https://thes.bncf.firenze.sbn.it/termine.php?id=46156">46156</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://id.loc.gov/authorities/subjects/sh85084441">sh85084441</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4169745-5">4169745-5</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Francia" title="Biblioteca nazionale di Francia">BNF</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="francese">FR</abbr></span>) <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb119444311">cb119444311</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb119444311">(data)</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr></span>) <a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007529311005171">987007529311005171</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_della_Dieta_nazionale_del_Giappone" title="Biblioteca della Dieta nazionale del Giappone">NDL</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="giapponese">JA</abbr></span>) <a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00567250">00567250</a></span></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; 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