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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">Teorema della mappa di Riemann</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 16 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-16" 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">16 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Riemannscher_Abbildungssatz" title="Riemannscher Abbildungssatz - tedesco" lang="de" hreflang="de" data-title="Riemannscher Abbildungssatz" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Riemann_mapping_theorem" title="Riemann mapping theorem - inglese" lang="en" hreflang="en" data-title="Riemann mapping theorem" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_de_representaci%C3%B3n_conforme_de_Riemann" title="Teorema de representación conforme de Riemann - spagnolo" lang="es" hreflang="es" data-title="Teorema de representación conforme de Riemann" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Riemannin_kuvauslause" title="Riemannin kuvauslause - finlandese" lang="fi" hreflang="fi" data-title="Riemannin kuvauslause" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_l%27application_conforme" title="Théorème de l'application conforme - francese" lang="fr" hreflang="fr" data-title="Théorème de l'application conforme" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%A9%D7%9C_%D7%A8%D7%99%D7%9E%D7%9F" title="משפט ההעתקה של רימן - ebraico" lang="he" hreflang="he" data-title="משפט ההעתקה של רימן" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E3%81%AE%E5%86%99%E5%83%8F%E5%AE%9A%E7%90%86" title="リーマンの写像定理 - giapponese" lang="ja" hreflang="ja" data-title="リーマンの写像定理" 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A6%AC%EB%A7%8C_%EC%82%AC%EC%83%81_%EC%A0%95%EB%A6%AC" title="리만 사상 정리 - coreano" lang="ko" hreflang="ko" data-title="리만 사상 정리" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Afbeeldingstelling_van_Riemann" title="Afbeeldingstelling van Riemann - olandese" lang="nl" hreflang="nl" data-title="Afbeeldingstelling van Riemann" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_do_mapeamento_conforme_de_Riemann" title="Teorema do mapeamento conforme de Riemann - portoghese" lang="pt" hreflang="pt" data-title="Teorema do mapeamento conforme de Riemann" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD%D0%B0_%D0%BE%D0%B1_%D0%BE%D1%82%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B8" title="Теорема Римана об отображении - russo" lang="ru" hreflang="ru" data-title="Теорема Римана об отображении" 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/Riemann_mapping_theorem" title="Riemann mapping theorem - Simple English" lang="en-simple" hreflang="en-simple" data-title="Riemann mapping theorem" 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-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Riemanns_avbildningssats" title="Riemanns avbildningssats - svedese" lang="sv" hreflang="sv" data-title="Riemanns avbildningssats" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Riemann_d%C3%B6n%C3%BC%C5%9F%C3%BCm_teoremi" title="Riemann dönüşüm teoremi - turco" lang="tr" hreflang="tr" data-title="Riemann dönüşüm teoremi" 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%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%A0%D1%96%D0%BC%D0%B0%D0%BD%D0%B0_%D0%BF%D1%80%D0%BE_%D0%B2%D1%96%D0%B4%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%BD%D1%8F" title="Теорема Рімана про відображення - ucraino" lang="uk" hreflang="uk" data-title="Теорема Рімана про відображення" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%BB%8E%E6%9B%BC%E6%98%A0%E5%B0%84%E5%AE%9A%E7%90%86" title="黎曼映射定理 - cinese" lang="zh" hreflang="zh" data-title="黎曼映射定理" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div 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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"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Georg_Friedrich_Bernhard_Riemann.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Georg_Friedrich_Bernhard_Riemann.jpeg/220px-Georg_Friedrich_Bernhard_Riemann.jpeg" decoding="async" width="220" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Georg_Friedrich_Bernhard_Riemann.jpeg/330px-Georg_Friedrich_Bernhard_Riemann.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Georg_Friedrich_Bernhard_Riemann.jpeg/440px-Georg_Friedrich_Bernhard_Riemann.jpeg 2x" data-file-width="903" data-file-height="986" /></a><figcaption>Il matematico <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>.</figcaption></figure> <p>In <a href="/wiki/Matematica" title="Matematica">matematica</a>, e più precisamente in <a href="/wiki/Analisi_complessa" title="Analisi complessa">analisi complessa</a>, il <b>teorema della mappa di Riemann</b> è un risultato importante riguardante alcuni <a href="/wiki/Insieme_aperto" title="Insieme aperto">insiemi aperti</a> del <a href="/wiki/Piano_complesso" title="Piano complesso">piano complesso</a>, che collega l'analisi complessa alla <a href="/wiki/Topologia" title="Topologia">topologia</a>. </p><p>Il teorema è un ingrediente fondamentale della dimostrazione del più generale <a href="/wiki/Teorema_di_uniformizzazione_di_Riemann" title="Teorema di uniformizzazione di Riemann">teorema di uniformizzazione di Riemann</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Enunciato">Enunciato</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_della_mappa_di_Riemann&veaction=edit&section=1" title="Modifica la sezione Enunciato" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_della_mappa_di_Riemann&action=edit&section=1" title="Edit section's source code: Enunciato"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sia <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/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> il disco aperto del piano complesso di raggio unitario </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=\{z\in \mathbb {C} \ |\ |z|<1\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=\{z\in \mathbb {C} \ |\ |z|<1\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a9a0626a75caf8e9b3f7ea384fc98cbdd761ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.052ex; height:2.843ex;" alt="{\displaystyle D=\{z\in \mathbb {C} \ |\ |z|<1\}.}"></span></dd></dl> <p>Il <b>teorema della mappa</b> di Riemann asserisce il fatto seguente. </p> <div style="float:center; width:85%; padding:15px; background: #f5f8ff; border: 1px solid blue; margin-left:8px; margin-right:8px;margin-bottom:15px; text-align:left"> <p>Ogni <a href="/wiki/Insieme_aperto" title="Insieme aperto">insieme aperto</a> <a href="/wiki/Spazio_semplicemente_connesso" title="Spazio semplicemente connesso">semplicemente connesso</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}"> <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> del <a href="/wiki/Piano_complesso" title="Piano complesso">piano complesso</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></span>, distinto 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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></span>, è <a href="/wiki/Biolomorfismo" class="mw-redirect" title="Biolomorfismo">biolomorfo</a> al disco aperto <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/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>. </p> </div> <div class="mw-heading mw-heading2"><h2 id="Generalità_dell'enunciato"><span id="Generalit.C3.A0_dell.27enunciato"></span>Generalità dell'enunciato</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_della_mappa_di_Riemann&veaction=edit&section=2" title="Modifica la sezione Generalità dell'enunciato" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_della_mappa_di_Riemann&action=edit&section=2" title="Edit section's source code: Generalità dell'enunciato"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Nessuna_ipotesi_sul_bordo">Nessuna ipotesi sul bordo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_della_mappa_di_Riemann&veaction=edit&section=3" title="Modifica la sezione Nessuna ipotesi sul bordo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_della_mappa_di_Riemann&action=edit&section=3" title="Edit section's source code: Nessuna ipotesi sul bordo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Flocke.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Flocke.PNG/220px-Flocke.PNG" decoding="async" width="220" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Flocke.PNG/330px-Flocke.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/f/f0/Flocke.PNG 2x" data-file-width="368" data-file-height="330" /></a><figcaption>La <a href="/wiki/Parte_interna" title="Parte interna">parte interna</a> del <a href="/wiki/Fiocco_di_neve_di_Koch" class="mw-redirect" title="Fiocco di neve di Koch">fiocco di neve di Koch</a> è biolomorfa al disco.</figcaption></figure> <p>Il fattore più sorprendente di questo enunciato è la sua ampia generalità. L'ipotesi di semplice connessione informalmente asserisce che l'aperto "non contiene buchi". Nessuna ulteriore ipotesi è fatta però sull'insieme, che ad esempio non deve essere necessariamente la <a href="/wiki/Parte_interna" title="Parte interna">parte interna</a> di un dominio con bordo regolare. Il teorema è quindi valido ad esempio 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/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 la parte interna di una <a href="/wiki/Ellisse" title="Ellisse">ellisse</a> (che ha bordo regolare), di un <a href="/wiki/Quadrato_(geometria)" class="mw-redirect" title="Quadrato (geometria)">quadrato</a> (che ha bordo regolare solo a tratti), oppure di un oggetto più complesso, come il <a href="/wiki/Fiocco_di_Koch" class="mw-redirect" title="Fiocco di Koch">fiocco di Koch</a> (il cui bordo non è una <a href="/wiki/Curva_(matematica)" title="Curva (matematica)">curva</a>, ma un più complicato <a href="/wiki/Frattale" title="Frattale">frattale</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Il_piano_complesso_è_escluso"><span id="Il_piano_complesso_.C3.A8_escluso"></span>Il piano complesso è escluso</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_della_mappa_di_Riemann&veaction=edit&section=4" title="Modifica la sezione Il piano complesso è escluso" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_della_mappa_di_Riemann&action=edit&section=4" title="Edit section's source code: Il piano complesso è escluso"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Il caso in cui <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 il piano complesso è escluso dall'enunciato: infatti il piano complesso ed il disco aperto non sono <a href="/wiki/Biolomorfismo" class="mw-redirect" title="Biolomorfismo">biolomorfi</a> (benché siano <a href="/wiki/Omeomorfismo" title="Omeomorfismo">omeomorfi</a>). Questo perché un biolomorfismo fornirebbe una <a href="/wiki/Funzione_intera" title="Funzione intera">funzione intera</a> <a href="/wiki/Funzione_limitata" title="Funzione limitata">limitata</a> ma non <a href="/wiki/Funzione_costante" title="Funzione costante">costante</a>, contraddicendo il <a href="/wiki/Teorema_di_Liouville_(analisi_complessa)" title="Teorema di Liouville (analisi complessa)">teorema di Liouville</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Conseguenze_topologiche">Conseguenze topologiche</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_della_mappa_di_Riemann&veaction=edit&section=5" title="Modifica la sezione Conseguenze topologiche" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_della_mappa_di_Riemann&action=edit&section=5" title="Edit section's source code: Conseguenze topologiche"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'enunciato ha una conseguenza <a href="/wiki/Topologia" title="Topologia">topologica</a> importante e di non ovvia dimostrazione: ogni sottoinsieme aperto semplicemente connesso del piano è <a href="/wiki/Omeomorfismo" title="Omeomorfismo">omeomorfo</a> al disco aperto<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>. </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=Teorema_della_mappa_di_Riemann&veaction=edit&section=6" 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=Teorema_della_mappa_di_Riemann&action=edit&section=6" 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-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"> Per l'enunciato topologico non è necessario escludere il caso in cui l'aperto sia tutto il piano, perché il piano è anch'esso omeomorfo al disco (benché non sia biolomorfo!)</span> </li> </ol></div> <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=Teorema_della_mappa_di_Riemann&veaction=edit&section=7" 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=Teorema_della_mappa_di_Riemann&action=edit&section=7" 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/Biolomorfismo" class="mw-redirect" title="Biolomorfismo">Biolomorfismo</a></li> <li><a href="/wiki/Teorema_di_uniformizzazione_di_Riemann" title="Teorema di uniformizzazione di Riemann">Teorema di uniformizzazione di Riemann</a></li> <li><a href="/wiki/Superficie_di_Riemann" title="Superficie di Riemann">Superficie di Riemann</a></li> <li><a href="/wiki/Lemma_di_Schwarz" title="Lemma di Schwarz">Lemma di Schwarz</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=Teorema_della_mappa_di_Riemann&veaction=edit&section=8" 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=Teorema_della_mappa_di_Riemann&action=edit&section=8" 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=""><span class="plainlinks" title="commons:Category:Riemann mapping"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Riemann_mapping?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><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:Riemann_mapping?uselang=it">teorema della mappa di Riemann</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=Teorema_della_mappa_di_Riemann&veaction=edit&section=9" 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=Teorema_della_mappa_di_Riemann&action=edit&section=9" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://planetmath.org/encyclopedia/ProofOfRiemannMappingTheorem.html">Una dimostrazione del teorema</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100620214524/http://planetmath.org/encyclopedia/ProofOfRiemannMappingTheorem.html">Archiviato</a> il 20 giugno 2010 in <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>., da <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a></li></ul> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Crystal128-kmplot.svg" class="mw-file-description" title="Matematica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/25px-Crystal128-kmplot.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/38px-Crystal128-kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/50px-Crystal128-kmplot.svg.png 2x" data-file-width="245" data-file-height="244" /></a></span> <b><a href="/wiki/Portale:Matematica" title="Portale:Matematica">Portale Matematica</a></b>: accedi alle voci di Wikipedia che trattano di matematica</div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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