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Disponibile in 15 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-15" 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">15 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/Weg_(Mathematik)" title="Weg (Mathematik) - tedesco" lang="de" hreflang="de" data-title="Weg (Mathematik)" 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/Path_(topology)" title="Path (topology) - inglese" lang="en" hreflang="en" data-title="Path (topology)" 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/Camino_(topolog%C3%ADa)" title="Camino (topología) - spagnolo" lang="es" hreflang="es" data-title="Camino (topología)" 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/Tee_(topoloogia)" title="Tee (topoloogia) - estone" lang="et" hreflang="et" data-title="Tee (topoloogia)" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Polku_(topologia)" title="Polku (topologia) - finlandese" lang="fi" hreflang="fi" data-title="Polku (topologia)" 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/Chemin_(topologie)" title="Chemin (topologie) - francese" lang="fr" hreflang="fr" data-title="Chemin (topologie)" 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%A1%D7%99%D7%9C%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="מסילה (מתמטיקה) - ebraico" lang="he" hreflang="he" data-title="מסילה (מתמטיקה)" 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/%E9%81%93_(%E4%BD%8D%E7%9B%B8%E5%B9%BE%E4%BD%95%E5%AD%A6)" title="道 (位相幾何学) - giapponese" lang="ja" hreflang="ja" data-title="道 (位相幾何学)" 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/%EA%B2%BD%EB%A1%9C_(%EC%9C%84%EC%83%81%EC%88%98%ED%95%99)" title="경로 (위상수학) - coreano" lang="ko" hreflang="ko" data-title="경로 (위상수학)" 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/Pad_(topologie)" title="Pad (topologie) - olandese" lang="nl" hreflang="nl" data-title="Pad (topologie)" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Droga_(topologia)" title="Droga (topologia) - polacco" lang="pl" hreflang="pl" data-title="Droga (topologia)" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Caminho_(topologia)" title="Caminho (topologia) - portoghese" lang="pt" hreflang="pt" data-title="Caminho (topologia)" 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%9F%D1%83%D1%82%D1%8C_(%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F)" title="Путь (топология) - russo" lang="ru" hreflang="ru" data-title="Путь (топология)" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A8%D0%BB%D1%8F%D1%85_(%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D1%96%D1%8F)" title="Шлях (топологія) - ucraino" lang="uk" hreflang="uk" data-title="Шлях (топологія)" 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%81%93%E8%B7%AF_(%E6%8B%93%E6%89%91%E5%AD%A6)" title="道路 (拓扑学) - cinese" lang="zh" hreflang="zh" data-title="道路 (拓扑学)" data-language-autonym="中文" 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href="https://www.wikidata.org/wiki/Special:EntityPage/Q1366002" title="Collegamento all'elemento connesso 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>In <a href="/wiki/Matematica" title="Matematica">matematica</a>, un <b>arco</b> (o <b>cammino</b>) in uno <a href="/wiki/Spazio_topologico" title="Spazio topologico">spazio topologico</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> è una <a href="/wiki/Funzione_continua" title="Funzione continua">funzione continua</a> dall'<a href="/wiki/Intervallo_(matematica)" title="Intervallo (matematica)">intervallo</a> unitario <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ec65159c44769434523e46928bc1b82681f842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.923ex; height:2.843ex;" alt="{\displaystyle I=[0,1]}"></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>. </p><p>Gli archi sono alla base della definizione del <a href="/wiki/Gruppo_fondamentale" title="Gruppo fondamentale">gruppo fondamentale</a>, e quindi della <a href="/wiki/Topologia_algebrica" title="Topologia algebrica">topologia algebrica</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definizioni">Definizioni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arco_(topologia)&veaction=edit&section=1" title="Modifica la sezione Definizioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Arco_(topologia)&action=edit&section=1" title="Edit section's source code: Definizioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Il <i>punto iniziale</i> dell'arco 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 f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc54adf96acf7e2466de00b7739144d36840108f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.25ex; height:2.843ex;" alt="{\displaystyle f(0)}"></span>, mentre il <i>punto finale</i> 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 f(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/948f00426d32f835b4cefa1a56aebd8e18ba7d3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.25ex; height:2.843ex;" alt="{\displaystyle f(1)}"></span>. 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> 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> sono due punti 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> (anche coincidenti), un "arco 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 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> 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 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>" è un arco 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> il cui punto iniziale 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 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 il cui punto finale 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 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>. 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=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/409a91214d63eabe46ec10ff3cbba689ab687366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}"></span>, si parla di <b>cammino chiuso</b>, o di <b>cappio</b> o di <b>laccio</b>. </p><p>Notiamo che un arco non è solamente un sottoinsieme 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>, ma una funzione 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 I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></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>: possono esistere archi diversi ma con lo stesso punto iniziale e finale e con la stessa immagine. </p><p>Uno spazio topologico in cui per ogni 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}"> <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 punti esiste un arco avente questi come punti iniziale e finale è detto <a href="/wiki/Spazio_connesso" title="Spazio connesso">connesso per archi</a>; poiché ogni punto è contenuto in un massimo sottospazio connesso per archi, ogni spazio topologico può essere decomposto in modo unico in componenti connesse per archi. L'insieme delle componenti connesse per archi 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> viene denotato con il simbolo <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 _{0}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{0}(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/674b6a9c0bfedc5b8137473d63973f7074f6d7b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.169ex; height:2.843ex;" alt="{\displaystyle \pi _{0}(X)}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Composizione">Composizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arco_(topologia)&veaction=edit&section=2" title="Modifica la sezione Composizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Arco_(topologia)&action=edit&section=2" title="Edit section's source code: Composizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Due archi <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> 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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></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/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> tali 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 f(1)=g(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1)=g(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9ad3022c1f42dfd1bbfacb00c7162106e77e44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.437ex; height:2.843ex;" alt="{\displaystyle f(1)=g(0)}"></span> possono essere composti, dando luogo ad un nuovo arco <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=f\ast g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=f\ast g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62376a817c814b19e95d82cfb8b9234a2165ed67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.027ex; height:2.509ex;" alt="{\displaystyle h=f\ast g}"></span>, che può essere visto come l'arco ottenuto percorrendo prima <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> e poi <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>: formalmente, <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/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> è la funzione 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 [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span> 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> 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 h(s)={\begin{cases}f(2s)&0\leq s\leq {\frac {1}{2}}\\g(2s-1)&{\frac {1}{2}}\leq s\leq 1.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> <mo>≤</mo> <mi>s</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>g</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>≤</mo> <mi>s</mi> <mo>≤</mo> <mn>1.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(s)={\begin{cases}f(2s)&0\leq s\leq {\frac {1}{2}}\\g(2s-1)&{\frac {1}{2}}\leq s\leq 1.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713db89c4dc072f4d475a41c7e10d74c027b5f73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.22ex; height:7.509ex;" alt="{\displaystyle h(s)={\begin{cases}f(2s)&0\leq s\leq {\frac {1}{2}}\\g(2s-1)&{\frac {1}{2}}\leq s\leq 1.\end{cases}}}"></span>.</dd></dl> <p>Questa operazione non è <a href="/wiki/Associativit%C3%A0" title="Associatività">associativa</a>: infatti, <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\ast g)\ast h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\ast g)\ast h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3287893831d5f8ce1c13afa8b7d81cd0223c48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.932ex; height:2.843ex;" alt="{\displaystyle (f\ast g)\ast h}"></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 f\ast (g\ast h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\ast (g\ast h)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c223d193080fa9a19f4aa7ad47f6ada399538a54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.932ex; height:2.843ex;" alt="{\displaystyle f\ast (g\ast h)}"></span> hanno lo stesso supporto, ma viaggiano a "velocità diverse": la prima percorre <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> in un tempo ¼, quindi <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> in un altro ¼, 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 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/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> in tempo 1/2; la seconda invece percorre <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> in tempo <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"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </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/e308a3a46b7fdce07cc09dcab9e8d8f73e37d935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 1/2}"></span> e le altre due ciascuna in tempo <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 1/4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3cf1ef33695c3d98cb09f01e5700f927ce928c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 1/4}"></span>. </p><p>Per risolvere questo problema si introduce la relazione di <a href="/wiki/Omotopia" title="Omotopia">equivalenza omotopica</a> tra archi, che permette tra l'altro di <a href="/wiki/Parametrizzazione" class="mw-redirect" title="Parametrizzazione">riparametrizzare</a> gli archi. L'insieme dei cammini chiusi con punto iniziale <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>, modulo omotopia, è chiamato <a href="/wiki/Gruppo_fondamentale" title="Gruppo fondamentale">gruppo fondamentale</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> con base <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>, ed è denotato 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="{\displaystyle \pi _{1}(X,x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}(X,x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c04f460bab6f580276cedce44809d2e466df4cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.586ex; height:2.843ex;" alt="{\displaystyle \pi _{1}(X,x_{0})}"></span>; 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 connesso per archi allora la <a href="/wiki/Isomorfismo_di_gruppi" class="mw-redirect" title="Isomorfismo di gruppi">classe di isomorfismo</a> di questo gruppo non dipende dal punto <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> scelto. </p> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arco_(topologia)&veaction=edit&section=3" 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=Arco_(topologia)&action=edit&section=3" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFserne" class="citation libro" style="font-style:normal"> Edoardo Sernesi, <span style="font-style:italic;">Geometria 2</span>, Torino, Bollati Boringhieri, 1994, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-339-5548-3" title="Speciale:RicercaISBN/978-88-339-5548-3">978-88-339-5548-3</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=Arco_(topologia)&veaction=edit&section=4" 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=Arco_(topologia)&action=edit&section=4" 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_connesso_per_archi" class="mw-redirect" title="Spazio connesso per archi">Spazio connesso per archi</a></li> <li><a href="/wiki/Spazio_semplicemente_connesso" title="Spazio semplicemente connesso">Spazio semplicemente connesso</a></li> <li><a href="/wiki/Omotopia" title="Omotopia">Omotopia</a></li> <li><a href="/wiki/Gruppo_fondamentale" title="Gruppo fondamentale">Gruppo fondamentale</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output .navbox_abovebelow{background:#ddf;font-size:90%;font-weight:normal}.mw-parser-output .navbox_group{background:#ddf;font-size:90%;padding:0 10px;white-space:nowrap}.mw-parser-output .navbox_list{font-size:90%;width:100%}.mw-parser-output .navbox_list a{white-space:nowrap}html:not(.vector-feature-night-mode-enabled) .mw-parser-output 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mw-collapsible mw-collapsed noprint metadata" id="navbox-Topologia"><tbody><tr><th colspan="3"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; 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 href="/wiki/Spazio_metrico" title="Spazio metrico">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 class="mw-selflink selflink">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 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