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Oppervlak (topologie) - Wikipedia
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dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <p>In de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Topologie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Topologie">topologie</a>, een deelgebied van de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Wiskunde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wiskunde">wiskunde</a>, is een <b>oppervlak</b> een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Dimensie_(algemeen)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dimensie (algemeen)">tweedimensionale</a> <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Topologische_vari%C3%ABteit?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Topologische variëteit">topologische variëteit</a>. De bekendste voorbeelden van oppervlakken zijn de begrenzingen van vaste <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Lichaam_(meetkunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lichaam (meetkunde)">lichamen</a> in de gewone driedimensionale <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Euclidische_ruimte?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidische ruimte">euclidische ruimte</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 {R} ^{3}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{3}.} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b00b2b4fd27c2cbffa02df568472f77b194a6db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}.}"></span> Aan de andere kant bestaan er oppervlakken die niet kunnen worden <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Inbedding?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Inbedding">ingebed</a> in de driedimensionale euclidische ruimte zonder <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Singulariteit_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Singulariteit (wiskunde)">singulariteiten</a> te introduceren of zonder dat deze oppervlakken zichzelf kruisen - dat zijn de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Ori%C3%ABntatie_(stand)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Oriëntatie (stand)">niet-oriënteerbare</a> oppervlakken. Op <i>oriënteerbare</i> oppervlakken kan men twee kanten aanwijzen, bijvoorbeeld de binnen- en buitenkant van een <a href="https://nl-m-wikipedia-org.translate.goog/w/index.php?title=Bal_(wiskunde)&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Bal (wiskunde) (de pagina bestaat niet)">bal</a>. Bij niet-oriënteerbare oppervlakken is dat niet mogelijk, een voorbeeld van een niet-oriënteerbare oppervlak is de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/M%C3%B6biusband?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Möbiusband">möbiusband</a>.</p> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Bestand:M%C3%B6bius_strip.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/M%C3%B6bius_strip.jpg/260px-M%C3%B6bius_strip.jpg" decoding="async" width="260" height="161" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/M%25C3%25B6bius_strip.jpg/390px-M%25C3%25B6bius_strip.jpg 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/M%25C3%25B6bius_strip.jpg/520px-M%25C3%25B6bius_strip.jpg 2x" data-file-width="1328" data-file-height="824"></a> <figcaption> De <a href="https://nl-m-wikipedia-org.translate.goog/wiki/M%C3%B6biusband?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Möbiusband">Möbiusband</a>: een glad, niet-oriënteerbaar oppervlak </figcaption> </figure> <p>Dat een oppervlak "tweedimensionaal" is, wil zeggen dat rondom elk <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Punt_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Punt (wiskunde)">punt</a> van het oppervlak een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Omgeving_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Omgeving (wiskunde)">omgeving</a> bestaat waarop een tweedimensionaal <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Co%C3%B6rdinatenstelsel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Coördinatenstelsel">coördinatensysteem</a> kan worden gedefinieerd. Het oppervlak van de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Aarde_(planeet)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aarde (planeet)">aarde</a> is bijvoorbeeld (idealiter) een tweedimensionale <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Sfeer_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sfeer (wiskunde)">sfeer</a>, waar de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Breedtegraad?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Breedtegraad">breedte-</a> en <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Lengtegraad?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lengtegraad">lengtegraad</a> de coördinaten zijn - behalve op de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Geografische_pool?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geografische pool">polen</a> en de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Internationale_datumgrens?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Internationale datumgrens">internationale datumgrens</a>, waar de lengtegraad niet gedefinieerd is. Dit voorbeeld illustreert dat een enkel coördinatensysteem niet voor alle oppervlakken volstaat. In het algemeen zijn er meerdere coördinatensystemen nodig om een oppervlak te <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Overdekking_(topologie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Overdekking (topologie)">overdekken</a>.</p> <p>Oppervlakken zijn onderwerp van studie in de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Natuurkunde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Natuurkunde">natuurkunde</a>, de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Techniek?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Techniek">techniek</a>, <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Computergraphics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Computergraphics">computergraphics</a> en vele andere disciplines, vooral wanneer zij de oppervlakken van fysieke objecten weergeven. In het analyseren van de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Aerodynamica?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aerodynamica">aerodynamische</a> eigenschappen van een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Vliegtuig?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vliegtuig">vliegtuig</a> is het centrale object van studie bijvoorbeeld de luchtstroom die langs het oppervlak van (de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Vleugel_(vliegtuig)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vleugel (vliegtuig)">vleugel</a> van) het vliegtuig loopt.</p> <p>Op een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Differentieerbaarheid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Differentieerbaarheid">glad</a> vlak is overal eenduidig een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Normaalvector?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Normaalvector">normaalvector</a> te definiëren. Een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Veelvlak?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Veelvlak">veelvlak</a> is vanwege de knikken geen glad vlak. Een bijzonder geval van een glad vlak is een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Vlak_(meetkunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vlak (meetkunde)">plat vlak</a>; de normaalvector heeft daar overal dezelfde richting.</p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="nl" dir="ltr"> <h2 id="mw-toc-heading">Inhoud</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://nl-m-wikipedia-org.translate.goog/wiki/Oppervlak_(topologie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Verbonden_sommen"><span class="tocnumber">1</span> <span class="toctext">Verbonden sommen</span></a></li> <li class="toclevel-1 tocsection-2"><a href="https://nl-m-wikipedia-org.translate.goog/wiki/Oppervlak_(topologie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Classificatie_van_gesloten_oppervlakken"><span class="tocnumber">2</span> <span class="toctext">Classificatie van gesloten oppervlakken</span></a></li> <li class="toclevel-1 tocsection-3"><a href="https://nl-m-wikipedia-org.translate.goog/wiki/Oppervlak_(topologie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Oppervlakken_in_de_meetkunde"><span class="tocnumber">3</span> <span class="toctext">Oppervlakken in de meetkunde</span></a></li> <li class="toclevel-1 tocsection-4"><a href="https://nl-m-wikipedia-org.translate.goog/wiki/Oppervlak_(topologie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Zie_ook"><span class="tocnumber">4</span> <span class="toctext">Zie ook</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Verbonden_sommen">Verbonden sommen</h2><span class="mw-editsection"> <a role="button" href="https://nl-m-wikipedia-org.translate.goog/w/index.php?title=Oppervlak_(topologie)&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bewerk dit kopje: Verbonden sommen" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>bewerken</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>De <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Verbonden_som?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Verbonden som">verbonden som</a> van twee oppervlakken <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> N </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle N,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2285a1804b7fdcac187d155af09aff63152dd56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.509ex;" alt="{\displaystyle N,}"> </noscript><span class="lazy-image-placeholder" style="width: 2.71ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2285a1804b7fdcac187d155af09aff63152dd56" data-alt="{\displaystyle N,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> aangeduid door <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\#N,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> N </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M\#N,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc58c7d55f99d7ad0e6221430bcc36da52e4d29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.089ex; height:2.509ex;" alt="{\displaystyle M\#N,}"> </noscript><span class="lazy-image-placeholder" style="width: 7.089ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc58c7d55f99d7ad0e6221430bcc36da52e4d29" data-alt="{\displaystyle M\#N,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> wordt verkregen door uit elk oppervlak een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Schijf_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Schijf (wiskunde)">schijf</a> te verwijderen en deze oppervlakken vervolgens langs de resulterende <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Rand_(topologie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rand (topologie)">begrenzing</a> aan elkaar te lijmen. De begrenzing van een schijf is een cirkel, zodat de begrenzende componenten hier cirkels zijn. De <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Euler-karakteristiek?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Euler-karakteristiek">euler-karakteristiek</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 \chi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> χ<!-- χ --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \chi } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }"> </noscript><span class="lazy-image-placeholder" style="width: 1.455ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" data-alt="{\displaystyle \chi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> van de verbonden som <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\#N}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> N </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M\#N} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7b0504c47f566ed4fdc6a6850e4538f20a62df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.442ex; height:2.509ex;" alt="{\displaystyle M\#N}"> </noscript><span class="lazy-image-placeholder" style="width: 6.442ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7b0504c47f566ed4fdc6a6850e4538f20a62df" data-alt="{\displaystyle M\#N}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is de som van de euler-karakteristieken van de summandi verminderd met twee:</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 \chi (M\#N)=\chi (M)+\chi (N)-2.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> χ<!-- χ --> </mi> <mo stretchy="false"> ( </mo> <mi> M </mi> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> N </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> χ<!-- χ --> </mi> <mo stretchy="false"> ( </mo> <mi> M </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> χ<!-- χ --> </mi> <mo stretchy="false"> ( </mo> <mi> N </mi> <mo stretchy="false"> ) </mo> <mo> −<!-- − --> </mo> <mn> 2. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \chi (M\#N)=\chi (M)+\chi (N)-2.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197e412f62badee306226fe4bc0338fc4d6a80d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.329ex; height:2.843ex;" alt="{\displaystyle \chi (M\#N)=\chi (M)+\chi (N)-2.}"> </noscript><span class="lazy-image-placeholder" style="width: 31.329ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197e412f62badee306226fe4bc0338fc4d6a80d7" data-alt="{\displaystyle \chi (M\#N)=\chi (M)+\chi (N)-2.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>De sfeer <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 S}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> </noscript><span class="lazy-image-placeholder" style="width: 1.499ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" data-alt="{\displaystyle S}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Neutraal_element?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Neutraal element">identiteitselement</a> voor de verbonden som, wat inhoudt dat <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 S\#M=M.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> M </mi> <mo> = </mo> <mi> M </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S\#M=M.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/303c294ab2166315b95458b0eb0ac0edb3e0ed03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.065ex; height:2.509ex;" alt="{\displaystyle S\#M=M.}"> </noscript><span class="lazy-image-placeholder" style="width: 12.065ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/303c294ab2166315b95458b0eb0ac0edb3e0ed03" data-alt="{\displaystyle S\#M=M.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> Dit is zo, omdat het verwijderen van een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Schijf_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Schijf (wiskunde)">schijf</a> uit de sfeer een schijf achterlaat, die de uit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> verwijderde schijf "upon" lijmen eenvoudig vervangt.</p> <p>Het uitvoeren van verbonden sommen met de torus <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 T}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> T </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> </noscript><span class="lazy-image-placeholder" style="width: 1.636ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" data-alt="{\displaystyle T}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> wordt ook wel beschreven als het aanbrengen van een "<a href="https://nl-m-wikipedia-org.translate.goog/wiki/Handvat_(topologie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Handvat (topologie)">handvat</a>" op de andere summand <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> Als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> oriënteerbaar is, dan is <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 T\#M=M.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> T </mi> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> M </mi> <mo> = </mo> <mi> M </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T\#M=M.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15089f884d1e77ffcc78cf1d9a8f53b8139ace0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.202ex; height:2.509ex;" alt="{\displaystyle T\#M=M.}"> </noscript><span class="lazy-image-placeholder" style="width: 12.202ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15089f884d1e77ffcc78cf1d9a8f53b8139ace0" data-alt="{\displaystyle T\#M=M.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> dit ook. De verbonden som is <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Associativiteit_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Associativiteit (wiskunde)">associatief</a>, zodat de verbonden som van een eindig aantal oppervlakken "goed gedefinieerd" is.</p> <p>De verbonden som van twee reële projectieve vlakken is de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Kleinfles?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Kleinfles">Klein-fles</a>. De verbonden som van het <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Re%C3%ABel_projectief_vlak?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Reëel projectief vlak">reële projectieve vlak</a> en de Klein-fles is <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Homeomorfisme?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Homeomorfisme">homeomorf</a> met de verbonden som van het reële projectieve vlak met de torus. De verbonden som van drie reële projectieve vlakken is dus homeomorf met de verbonden som van het reële projectieve vlak met de torus. Elke verbonden som, waar een reëel projectief deel van uitmaakt, is niet-oriënteerbaar.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Classificatie_van_gesloten_oppervlakken">Classificatie van gesloten oppervlakken</h2><span class="mw-editsection"> <a role="button" href="https://nl-m-wikipedia-org.translate.goog/w/index.php?title=Oppervlak_(topologie)&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bewerk dit kopje: Classificatie van gesloten oppervlakken" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>bewerken</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>De <i>classificatiestelling van gesloten oppervlakken</i> stelt dat elk gesloten oppervlak homeomorf is met een bepaald lid van een van drie onderstaande families:</p> <ol> <li>De <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Sfeer_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sfeer (wiskunde)">sfeer</a>;</li> <li>De <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Verbonden_som?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Verbonden som">verbonden som</a> van <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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.116ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" data-alt="{\displaystyle g}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Torus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Torus">torussen</a>, voor <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\geq 1;}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> g </mi> <mo> ≥<!-- ≥ --> </mo> <mn> 1 </mn> <mo> ; </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle g\geq 1;} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/052545c4ab62463c0760de36adc965ee32aced6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.024ex; height:2.509ex;" alt="{\displaystyle g\geq 1;}"> </noscript><span class="lazy-image-placeholder" style="width: 6.024ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/052545c4ab62463c0760de36adc965ee32aced6c" data-alt="{\displaystyle g\geq 1;}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></li> <li>De verbonden som van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> </noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> reële <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Projectie_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Projectie (wiskunde)">projectieve vlakken</a>, voor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> <mo> ≥<!-- ≥ --> </mo> <mn> 1. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k\geq 1.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a1157fbb720faad2aadf78cbf780e21d3e9e38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.119ex; height:2.343ex;" alt="{\displaystyle k\geq 1.}"> </noscript><span class="lazy-image-placeholder" style="width: 6.119ex;height: 2.343ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a1157fbb720faad2aadf78cbf780e21d3e9e38" data-alt="{\displaystyle k\geq 1.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></li> </ol> <p>De oppervlakken in de eerste twee families zijn <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Ori%C3%ABnteerbaarheid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Oriënteerbaarheid">oriënteerbaar</a>. Het is handig om de twee families te combineren door een sfeer op te vatten als de verbonden som van 0 torussen. Het aantal <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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.116ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" data-alt="{\displaystyle g}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> betrokken torussen noemt men het <i><a href="https://nl-m-wikipedia-org.translate.goog/wiki/Genus_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Genus (wiskunde)">genus</a></i> van het oppervlak. Aangezien de sfeer en de torus een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Euler-karakteristiek?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Euler-karakteristiek">euler-karakteristiek</a> van respectievelijk 2 en 0 hebben, volgt hieruit dat de euler-karakteristiek van de verbonden som van <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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.116ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" data-alt="{\displaystyle g}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> torussen gelijk is aan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2-2g.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 2 </mn> <mo> −<!-- − --> </mo> <mn> 2 </mn> <mi> g </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 2-2g.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c1812ce845104f48816fe2c08d38fa10dc185a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.928ex; height:2.509ex;" alt="{\displaystyle 2-2g.}"> </noscript><span class="lazy-image-placeholder" style="width: 6.928ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c1812ce845104f48816fe2c08d38fa10dc185a" data-alt="{\displaystyle 2-2g.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>De oppervlakken in de derde familie zijn niet-oriënteerbaar. Aangezien de euler-karakteristiek van het reële projectieve vlak gelijk is aan 1, is de euler-karakteristiek van de verbonden som van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> </noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> oppervlakken gelijk aan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2-k.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 2 </mn> <mo> −<!-- − --> </mo> <mi> k </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 2-k.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c5c08e1068003d694dab343966a5a6ebbc7179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.861ex; height:2.343ex;" alt="{\displaystyle 2-k.}"> </noscript><span class="lazy-image-placeholder" style="width: 5.861ex;height: 2.343ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c5c08e1068003d694dab343966a5a6ebbc7179" data-alt="{\displaystyle 2-k.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>Hieruit volgt dat een gesloten oppervlak, "up to" <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Homeomorfisme?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Homeomorfisme">homeomorfisme</a>, wordt bepaald door twee "stukjes" <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Informatie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Informatie">informatie</a>: haar <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Euler-karakteristiek?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Euler-karakteristiek">euler-karakteristiek</a> en of het oppervlak <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Ori%C3%ABnteerbaarheid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Oriënteerbaarheid">oriënteerbaar</a> is of niet. Met andere woorden de euler-karakteristiek en de oriënteerbaarheid geven een volledige classificatie van gesloten oppervlakken "up to" homeomorfisme.</p> <p>Deze classificatie in relatie brengend met de verbonden som, vormen de gesloten oppervlakken "up to" homeomorfisme een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Mono%C3%AFde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Monoïde">monoïde</a> met betrekking tot de verbonden som. De identiteit is de sfeer. Het reële projectieve vlak en de torus genereren deze monoïde. In aanvulling hierop is er een relatie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\#P\#P=p\#T}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> P </mi> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> P </mi> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> P </mi> <mo> = </mo> <mi> p </mi> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> T </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P\#P\#P=p\#T} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/801ca6e6e5cbe82fca93d561d9f630362e2b8f75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.948ex; height:2.509ex;" alt="{\displaystyle P\#P\#P=p\#T}"> </noscript><span class="lazy-image-placeholder" style="width: 16.948ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/801ca6e6e5cbe82fca93d561d9f630362e2b8f75" data-alt="{\displaystyle P\#P\#P=p\#T}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> - meetkundig, verbonden som met een torus (<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 \#T}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> T </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \#T} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a00e4bde1ba8623176c03e277808a19a57ef82d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.572ex; height:2.509ex;" alt="{\displaystyle \#T}"> </noscript><span class="lazy-image-placeholder" style="width: 3.572ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a00e4bde1ba8623176c03e277808a19a57ef82d" data-alt="{\displaystyle \#T}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>) voegt een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Handvat_(topologie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Handvat (topologie)">handvat</a> toe met beide uiteinden vastgemaakt aan dezelfde <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Zijde_(meetkunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Zijde (meetkunde)">zijde</a> van het oppervlak, terwijl de verbonden som met een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Kleinfles?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Kleinfles">Klein-fles</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 \#K=\#P\#P}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> K </mi> <mo> = </mo> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> P </mi> <mi mathvariant="normal"> #<!-- # --> </mi> <mi> P </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \#K=\#P\#P} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/981601254ce7987a35179d481adf6ccbdf9d50c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.463ex; height:2.509ex;" alt="{\displaystyle \#K=\#P\#P}"> </noscript><span class="lazy-image-placeholder" style="width: 14.463ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/981601254ce7987a35179d481adf6ccbdf9d50c8" data-alt="{\displaystyle \#K=\#P\#P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>) een handvat met de twee uiteinden vastgemaakt aan weerszijden van het oppervlak, in de aanwezigheid van een projectief vlak, is het oppervlak niet oriënteerbaar (er is geen notie van een zijde), dus er is geen verschil tussen het vastmaken van een torus en het vastmaken van een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Kleinfles?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Kleinfles">Klein-fles</a>, wat de relatie verklaart.</p> <p>Er bestaan een aantal <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Wiskundig_bewijs?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wiskundig bewijs">bewijzen</a> van deze classificatie; het meest gebruikelijke bewijs steunt op het moeilijke resultaat dat elke <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Compact?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Compact">compacte</a> <a href="https://nl-m-wikipedia-org.translate.goog/w/index.php?title=2-vari%C3%ABteit&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="2-variëteit (de pagina bestaat niet)">2-variëteit</a> <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Homeomorfisme?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Homeomorfisme">homeomorf</a> is aan een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Simpliciaal_complex?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Simpliciaal complex">simpliciaal complex</a>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Oppervlakken_in_de_meetkunde">Oppervlakken in de meetkunde</h2><span class="mw-editsection"> <a role="button" href="https://nl-m-wikipedia-org.translate.goog/w/index.php?title=Oppervlak_(topologie)&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bewerk dit kopje: Oppervlakken in de meetkunde" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>bewerken</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <div class="hatnote" style="margin-bottom:0.5em; padding:0.5em 0 0.5em 1.6em; font-size:95%;" role="note"> <span typeof="mw:File"><span> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/15px-1rightarrow_blue.svg.png" decoding="async" width="15" height="15" class="mw-file-element" data-file-width="480" data-file-height="480"> </noscript><span class="lazy-image-placeholder" style="width: 15px;height: 15px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/15px-1rightarrow_blue.svg.png" data-alt="" data-width="15" data-height="15" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/23px-1rightarrow_blue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/30px-1rightarrow_blue.svg.png 2x" data-class="mw-file-element"> </span></span></span> <i>Zie <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Differentiaalmeetkunde_van_oppervlakken?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Differentiaalmeetkunde van oppervlakken">Differentiaalmeetkunde van oppervlakken</a> voor het hoofdartikel over dit onderwerp.</i> </div> <p><a href="https://nl-m-wikipedia-org.translate.goog/wiki/Veelvlak?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Veelvlak">Veelvlakken</a>, zoals de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Rand_(topologie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rand (topologie)">begrenzing</a> van een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Kubus_(ruimtelijke_figuur)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kubus (ruimtelijke figuur)">kubus</a>, behoren tot de eerste oppervlakken die men in de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Meetkunde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Meetkunde">meetkunde</a> tegenkomt. Het is ook mogelijk om <i>gladde oppervlakken</i> te definiëren, waarin elk <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Punt_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Punt (wiskunde)">punt</a> een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Omgeving_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Omgeving (wiskunde)">omgeving</a> heeft, die <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Diffeomorfisme?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Diffeomorfisme">diffeomorf</a> is aan enige <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Open_verzameling?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Open verzameling">open verzameling</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} ^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> E </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathrm {E} ^{2}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc52f6fda0f2a52c25fccf71e6f5361ac533b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.284ex; height:2.676ex;" alt="{\displaystyle \mathrm {E} ^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 3.284ex;height: 2.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc52f6fda0f2a52c25fccf71e6f5361ac533b61" data-alt="{\displaystyle \mathrm {E} ^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> Deze uitwerking laat toe dat <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Analyse_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Analyse (wiskunde)">analyse</a> kan worden toegepast op oppervlakken en dat zo veel resultaten <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Wiskundig_bewijs?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wiskundig bewijs">bewezen</a> kunnen worden.</p> <p>Twee gladde oppervlakken zijn <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Dan_en_slechts_dan_als?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dan en slechts dan als">dan en slechts dan</a> diffeomorf als zij <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Homeomorfisme?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Homeomorfisme">homeomorf</a> zijn. (Het analoge resultaat is niet van toepassing op hoger-dimensionale variëteiten). <a href="https://nl-m-wikipedia-org.translate.goog/w/index.php?title=Gesloten_oppervlak&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Gesloten oppervlak (de pagina bestaat niet)">Gesloten oppervlakken</a> zijn dus door hun <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Euler-karakteristiek?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Euler-karakteristiek">euler-karakteristiek</a> en <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Ori%C3%ABnteerbaarheid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Oriënteerbaarheid">oriënteerbaarheid</a> geclassificeerd "up to" diffeomorfisme.</p> <p>Gladde oppervlakken uitgerust met <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Riemann-metriek?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Riemann-metriek">Riemann-metrieken</a> zijn van fundamenteel belang in de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Differentiaalmeetkunde?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Differentiaalmeetkunde">differentiaalmeetkunde</a>. Een Riemann-metriek rust een oppervlak uit met een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Geodeet_(landmeetkunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geodeet (landmeetkunde)">geodetische</a> notie en een begrip van <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Afstand?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Afstand">afstanden</a>, <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Hoek_(meetkunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hoek (meetkunde)">hoeken</a> en <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Oppervlakte?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Oppervlakte">oppervlakten</a>. De Riemann-metriek geeft ook aanleiding tot <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Gaussiaanse_kromming?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gaussiaanse kromming">Gaussiaanse kromming</a>, die beschrijft hoe gekromd of gebogen het oppervlak op elk punt is. <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Kromming_(meetkunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kromming (meetkunde)">kromming</a> is een rigide, meetkunde eigenschap, in de zijn dat kromming niet wordt bewaard door algemene <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Diffeomorfisme?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Diffeomorfisme">diffeomorfismen</a> van het oppervlak. De beroemde <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Stelling_van_Gauss-Bonnet?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Stelling van Gauss-Bonnet">stelling van Gauss-Bonnet</a> voor gesloten oppervlakken stelt echter dat de integraal van de Gaussiaanse kromming <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> K </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle K} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> </noscript><span class="lazy-image-placeholder" style="width: 2.066ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" data-alt="{\displaystyle K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> over het gehele oppervlak <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 S}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> </noscript><span class="lazy-image-placeholder" style="width: 1.499ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" data-alt="{\displaystyle S}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> wordt bepaald door de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Euler-karakteristiek?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Euler-karakteristiek">euler-karakteristiek</a>:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{S}K\;\mathrm {d} A=2\pi \chi (S).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫<!-- ∫ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> S </mi> </mrow> </msub> <mi> K </mi> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> <mo> = </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> <mi> χ<!-- χ --> </mi> <mo stretchy="false"> ( </mo> <mi> S </mi> <mo stretchy="false"> ) </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{S}K\;\mathrm {d} A=2\pi \chi (S).} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6cb176ebca61e28b1080948ebb749d33be372d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.722ex; height:5.676ex;" alt="{\displaystyle \int _{S}K\;\mathrm {d} A=2\pi \chi (S).}"> </noscript><span class="lazy-image-placeholder" style="width: 19.722ex;height: 5.676ex;vertical-align: -2.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6cb176ebca61e28b1080948ebb749d33be372d3" data-alt="{\displaystyle \int _{S}K\;\mathrm {d} A=2\pi \chi (S).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Dit resultaat illustreert de diepe relatie tussen de meetkunde en topologie van oppervlakken (en, in mindere mate, hoger-dimensionale variëteiten).</p> <p>Een andere manier, waarop oppervlakken in de meetkunde ontstaan is door over te gaan naar het complexe domein. Een complexe <a href="https://nl-m-wikipedia-org.translate.goog/w/index.php?title=1-vari%C3%ABteit&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="1-variëteit (de pagina bestaat niet)">1-variëteit</a> is een glad georiënteerd oppervlak, ook wel een <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Riemann-oppervlak?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Riemann-oppervlak">Riemann-oppervlak</a> genoemd. Elke complexe niet-singuliere <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Algebra%C3%AFsche_kromme?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Algebraïsche kromme">algebraïsche kromme</a>, die gezien wordt als een reële variëteit, is een Riemann-oppervlak.</p> <p>Elk gesloten oriënteerbaar oppervlak laat een complexe <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Wiskundige_structuur?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wiskundige structuur">structuur</a> toe. Complexe structuren op een gesloten georiënteerd oppervlak komen overeen met <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Hoekgetrouwe_equivalentie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hoekgetrouwe equivalentie">hoekgetrouwe equivalentieklassen</a> van een Riemaniaanse-metriek op het oppervlak. Een versie van de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Uniformeringsstelling?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uniformeringsstelling">uniformeringsstelling</a> (door <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Henri_Poincar%C3%A9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Henri Poincaré">Poincaré</a>) stelt dat elke <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Riemann-vari%C3%ABteit?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Riemann-variëteit">Riemanniaanse metriek</a> op een oriënteerbare, gesloten oppervlak hoekgetrouw gelijkwaardig is aan een in wezen unieke metriek van <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Constante_kromming?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Constante kromming">constante kromming</a>. Dit biedt een uitgangspunt voor een van de benaderingen van de <a href="https://nl-m-wikipedia-org.translate.goog/w/index.php?title=Teichm%C3%BCller-theorie&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Teichmüller-theorie (de pagina bestaat niet)">Teichmüller-theorie</a>, die in een fijnmazigere classificatie van Riemann-oppervlakken voorziet als alleen de topologische classificatie door euler-karakteristieken.</p> <p>Een <i>complex oppervlak</i> is een complexe <a href="https://nl-m-wikipedia-org.translate.goog/w/index.php?title=2-vari%C3%ABteit&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="2-variëteit (de pagina bestaat niet)">2-variëteit</a> en dus een reële <a href="https://nl-m-wikipedia-org.translate.goog/wiki/4-vari%C3%ABteit?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="4-variëteit">4-variëteit</a>; in de zin van dit artikel is een <i>complex oppervlak</i> geen oppervlak. Noch zijn er algebraïsche krommen of algebraïsche oppervlakken over <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Veld_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Veld (wiskunde)">velden</a> gedefinieerd anders dan de <a href="https://nl-m-wikipedia-org.translate.goog/wiki/Complex_getal?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Complex getal">complexe getallen</a>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Zie_ook">Zie ook</h2><span class="mw-editsection"> <a role="button" href="https://nl-m-wikipedia-org.translate.goog/w/index.php?title=Oppervlak_(topologie)&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bewerk dit kopje: Zie ook" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>bewerken</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <ul> <li><a href="https://nl-m-wikipedia-org.translate.goog/wiki/Genus_(wiskunde)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Genus (wiskunde)">Genus (wiskunde)</a></li> </ul><!-- NewPP limit report Parsed by mw‐api‐ext.eqiad.main‐58bb4f8ccf‐lvskc Cached time: 20250214135916 Cache expiry: 2592000 Reduced expiry: false Complications: [show‐toc] CPU time usage: 0.085 seconds Real time usage: 0.422 seconds Preprocessor visited node count: 334/1000000 Post‐expand include size: 640/2097152 bytes Template argument size: 117/2097152 bytes Highest expansion depth: 9/100 Expensive parser function count: 0/500 Unstrip recursion depth: 0/20 Unstrip post‐expand size: 1080/5000000 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 5.429 1 -total 100.00% 5.429 1 Sjabloon:Zie_hoofdartikel 53.03% 2.879 1 Sjabloon:Multiparameters --> <!-- Saved in parser cache with key nlwiki:pcache:1663939:|#|:idhash:canonical and timestamp 20250214135916 and revision id 51087239. Rendering was triggered because: api-parse --> </section> </div><!-- MobileFormatter took 0.011 seconds --><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --> <noscript> <img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=mobile&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> Overgenomen van "<a dir="ltr" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/w/index.php?title%3DOppervlak_(topologie)%26oldid%3D51087239">https://nl.wikipedia.org/w/index.php?title=Oppervlak_(topologie)&oldid=51087239</a>" </div> </div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"><a class="last-modified-bar" href="https://nl-m-wikipedia-org.translate.goog/w/index.php?title=Oppervlak_(topologie)&action=history&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"><span class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="Madyno" data-user-gender="unknown" data-timestamp="1520014079"> <span>Laatst bewerkt op 2 mrt 2018, om 19:07</span> </span> <span class="minerva-icon minerva-icon-size-small minerva-icon--expand"></span> </div></a> <div class="post-content footer-content"> <div id="mw-data-after-content"> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Talen</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://af.wikipedia.org/wiki/Oppervlak" title="Oppervlak – Afrikaans" lang="af" hreflang="af" data-title="Oppervlak" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://als.wikipedia.org/wiki/Fl%25C3%25A4che_(Topologie)" title="Fläche (Topologie) – Zwitserduits" lang="gsw" hreflang="gsw" data-title="Fläche (Topologie)" data-language-autonym="Alemannisch" data-language-local-name="Zwitserduits" class="interlanguage-link-target"><span>Alemannisch</span></a></li> <li class="interlanguage-link interwiki-an mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://an.wikipedia.org/wiki/Superficie" title="Superficie – Aragonees" lang="an" hreflang="an" data-title="Superficie" data-language-autonym="Aragonés" data-language-local-name="Aragonees" class="interlanguage-link-target"><span>Aragonés</span></a></li> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ar.wikipedia.org/wiki/%25D8%25B3%25D8%25B7%25D8%25AD" title="سطح – Arabisch" lang="ar" hreflang="ar" data-title="سطح" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li> <li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ast.wikipedia.org/wiki/Superficie" title="Superficie – Asturisch" lang="ast" hreflang="ast" data-title="Superficie" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></a></li> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://az.wikipedia.org/wiki/S%25C9%2599th" title="Səth – Azerbeidzjaans" lang="az" hreflang="az" data-title="Səth" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbeidzjaans" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be.wikipedia.org/wiki/%25D0%259F%25D0%25B0%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D1%258F" title="Паверхня – Belarussisch" lang="be" hreflang="be" data-title="Паверхня" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></a></li> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bg.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D1%258A%25D1%2580%25D1%2585%25D0%25BD%25D0%25BE%25D1%2581%25D1%2582" title="Повърхност – Bulgaars" lang="bg" hreflang="bg" data-title="Повърхност" data-language-autonym="Български" data-language-local-name="Bulgaars" class="interlanguage-link-target"><span>Български</span></a></li> <li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bn.wikipedia.org/wiki/%25E0%25A6%25A4%25E0%25A6%25B2_(%25E0%25A6%259F%25E0%25A6%25AA%25E0%25A7%258B%25E0%25A6%25B2%25E0%25A6%259C%25E0%25A6%25BF)" title="তল (টপোলজি) – Bengaals" lang="bn" hreflang="bn" data-title="তল (টপোলজি)" data-language-autonym="বাংলা" data-language-local-name="Bengaals" class="interlanguage-link-target"><span>বাংলা</span></a></li> <li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bs.wikipedia.org/wiki/Povr%25C5%25A1" title="Površ – Bosnisch" lang="bs" hreflang="bs" data-title="Površ" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Superf%25C3%25ADcie_(matem%25C3%25A0tiques)" title="Superfície (matemàtiques) – Catalaans" lang="ca" hreflang="ca" data-title="Superfície (matemàtiques)" data-language-autonym="Català" data-language-local-name="Catalaans" class="interlanguage-link-target"><span>Català</span></a></li> <li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ckb.wikipedia.org/wiki/%25DA%2595%25D9%2588%25D9%2588" title="ڕوو – Soranî" lang="ckb" hreflang="ckb" data-title="ڕوو" data-language-autonym="کوردی" data-language-local-name="Soranî" class="interlanguage-link-target"><span>کوردی</span></a></li> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/Plocha" title="Plocha – Tsjechisch" lang="cs" hreflang="cs" data-title="Plocha" data-language-autonym="Čeština" data-language-local-name="Tsjechisch" class="interlanguage-link-target"><span>Čeština</span></a></li> <li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cv.wikipedia.org/wiki/%25C3%2587%25D0%25B8%25D0%25B9" title="Çий – Tsjoevasjisch" lang="cv" hreflang="cv" data-title="Çий" data-language-autonym="Чӑвашла" data-language-local-name="Tsjoevasjisch" class="interlanguage-link-target"><span>Чӑвашла</span></a></li> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Surface_(topology)" title="Surface (topology) – Engels" lang="en" hreflang="en" data-title="Surface (topology)" data-language-autonym="English" data-language-local-name="Engels" class="interlanguage-link-target"><span>English</span></a></li> <li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eo.wikipedia.org/wiki/Surfaco" title="Surfaco – Esperanto" lang="eo" hreflang="eo" data-title="Surfaco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Superficie_(topolog%25C3%25ADa)" title="Superficie (topología) – Spaans" lang="es" hreflang="es" data-title="Superficie (topología)" data-language-autonym="Español" data-language-local-name="Spaans" class="interlanguage-link-target"><span>Español</span></a></li> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/Pind" title="Pind – Estisch" lang="et" hreflang="et" data-title="Pind" data-language-autonym="Eesti" data-language-local-name="Estisch" class="interlanguage-link-target"><span>Eesti</span></a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Gainazal" title="Gainazal – Baskisch" lang="eu" hreflang="eu" data-title="Gainazal" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25D8%25B1%25D9%2588%25DB%258C%25D9%2587_(%25D8%25AA%25D9%2588%25D9%25BE%25D9%2588%25D9%2584%25D9%2588%25DA%2598%25DB%258C)" title="رویه (توپولوژی) – Perzisch" lang="fa" hreflang="fa" data-title="رویه (توپولوژی)" data-language-autonym="فارسی" data-language-local-name="Perzisch" class="interlanguage-link-target"><span>فارسی</span></a></li> <li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Pinta_(geometria)" title="Pinta (geometria) – Fins" lang="fi" hreflang="fi" data-title="Pinta (geometria)" data-language-autonym="Suomi" data-language-local-name="Fins" class="interlanguage-link-target"><span>Suomi</span></a></li> <li class="interlanguage-link interwiki-fr badge-Q70893996 mw-list-item" title=""><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Surface_(g%25C3%25A9om%25C3%25A9trie)" title="Surface (géométrie) – Frans" lang="fr" hreflang="fr" data-title="Surface (géométrie)" data-language-autonym="Français" data-language-local-name="Frans" class="interlanguage-link-target"><span>Français</span></a></li> <li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fur.wikipedia.org/wiki/Superficie" title="Superficie – Friulisch" lang="fur" hreflang="fur" data-title="Superficie" data-language-autonym="Furlan" data-language-local-name="Friulisch" class="interlanguage-link-target"><span>Furlan</span></a></li> <li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ga.wikipedia.org/wiki/Dromchla_(toipeola%25C3%25ADocht)" title="Dromchla (toipeolaíocht) – Iers" lang="ga" hreflang="ga" data-title="Dromchla (toipeolaíocht)" data-language-autonym="Gaeilge" data-language-local-name="Iers" class="interlanguage-link-target"><span>Gaeilge</span></a></li> <li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gl.wikipedia.org/wiki/Superficie" title="Superficie – Galicisch" lang="gl" hreflang="gl" data-title="Superficie" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%259E%25D7%25A9%25D7%2598%25D7%2597_(%25D7%2598%25D7%2595%25D7%25A4%25D7%2595%25D7%259C%25D7%2595%25D7%2592%25D7%2599%25D7%2594)" title="משטח (טופולוגיה) – Hebreeuws" lang="he" hreflang="he" data-title="משטח (טופולוגיה)" data-language-autonym="עברית" data-language-local-name="Hebreeuws" class="interlanguage-link-target"><span>עברית</span></a></li> <li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%25AA%25E0%25A5%2583%25E0%25A4%25B7%25E0%25A5%258D%25E0%25A4%259F" title="पृष्ट – Hindi" lang="hi" hreflang="hi" data-title="पृष्ट" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li> <li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hr.wikipedia.org/wiki/Ploha_(geometrija)" title="Ploha (geometrija) – Kroatisch" lang="hr" hreflang="hr" data-title="Ploha (geometrija)" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target"><span>Hrvatski</span></a></li> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/Felsz%25C3%25ADn" title="Felszín – Hongaars" lang="hu" hreflang="hu" data-title="Felszín" data-language-autonym="Magyar" data-language-local-name="Hongaars" class="interlanguage-link-target"><span>Magyar</span></a></li> <li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hy.wikipedia.org/wiki/%25D5%2584%25D5%25A1%25D5%25AF%25D5%25A5%25D6%2580%25D6%2587%25D5%25B8%25D6%2582%25D5%25B5%25D5%25A9" title="Մակերևույթ – Armeens" lang="hy" hreflang="hy" data-title="Մակերևույթ" data-language-autonym="Հայերեն" data-language-local-name="Armeens" class="interlanguage-link-target"><span>Հայերեն</span></a></li> <li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ia.wikipedia.org/wiki/Superficie" title="Superficie – Interlingua" lang="ia" hreflang="ia" data-title="Superficie" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li> <li class="interlanguage-link interwiki-id mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://id.wikipedia.org/wiki/Permukaan_(topologi)" title="Permukaan (topologi) – Indonesisch" lang="id" hreflang="id" data-title="Permukaan (topologi)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li> <li class="interlanguage-link interwiki-inh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://inh.wikipedia.org/wiki/%25D0%25A2%25D3%2580%25D0%25B5%25D1%2585%25D0%25B5" title="ТӀехе – Ingoesjetisch" lang="inh" hreflang="inh" data-title="ТӀехе" data-language-autonym="ГӀалгӀай" data-language-local-name="Ingoesjetisch" class="interlanguage-link-target"><span>ГӀалгӀай</span></a></li> <li class="interlanguage-link interwiki-io mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://io.wikipedia.org/wiki/Surfaco" title="Surfaco – Ido" lang="io" hreflang="io" data-title="Surfaco" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li> <li class="interlanguage-link interwiki-is mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://is.wikipedia.org/wiki/Yfirbor%25C3%25B0" title="Yfirborð – IJslands" lang="is" hreflang="is" data-title="Yfirborð" data-language-autonym="Íslenska" data-language-local-name="IJslands" class="interlanguage-link-target"><span>Íslenska</span></a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Superficie" title="Superficie – Italiaans" lang="it" hreflang="it" data-title="Superficie" data-language-autonym="Italiano" data-language-local-name="Italiaans" class="interlanguage-link-target"><span>Italiano</span></a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E6%259B%25B2%25E9%259D%25A2" title="曲面 – Japans" lang="ja" hreflang="ja" data-title="曲面" data-language-autonym="日本語" data-language-local-name="Japans" class="interlanguage-link-target"><span>日本語</span></a></li> <li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://kk.wikipedia.org/wiki/%25D0%2591%25D0%25B5%25D1%2582_(%25D0%25B3%25D0%25B5%25D0%25BE%25D0%25BC%25D0%25B5%25D1%2582%25D1%2580%25D0%25B8%25D1%258F)" title="Бет (геометрия) – Kazachs" lang="kk" hreflang="kk" data-title="Бет (геометрия)" data-language-autonym="Қазақша" data-language-local-name="Kazachs" class="interlanguage-link-target"><span>Қазақша</span></a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25EA%25B3%25A1%25EB%25A9%25B4" title="곡면 – Koreaans" lang="ko" hreflang="ko" data-title="곡면" data-language-autonym="한국어" data-language-local-name="Koreaans" class="interlanguage-link-target"><span>한국어</span></a></li> <li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ky.wikipedia.org/wiki/%25D0%2591%25D0%25B5%25D1%2582_(%25D0%2593%25D0%25B5%25D0%25BE%25D0%25BC%25D0%25B5%25D1%2582%25D1%2580%25D0%25B8%25D1%258F)" title="Бет (Геометрия) – Kirgizisch" lang="ky" hreflang="ky" data-title="Бет (Геометрия)" data-language-autonym="Кыргызча" data-language-local-name="Kirgizisch" class="interlanguage-link-target"><span>Кыргызча</span></a></li> <li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lij.wikipedia.org/wiki/Superfi%25C3%25A7ie_(matematica)" title="Superfiçie (matematica) – Ligurisch" lang="lij" hreflang="lij" data-title="Superfiçie (matematica)" data-language-autonym="Ligure" data-language-local-name="Ligurisch" class="interlanguage-link-target"><span>Ligure</span></a></li> <li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lt.wikipedia.org/wiki/Pavir%25C5%25A1ius" title="Paviršius – Litouws" lang="lt" hreflang="lt" data-title="Paviršius" data-language-autonym="Lietuvių" data-language-local-name="Litouws" class="interlanguage-link-target"><span>Lietuvių</span></a></li> <li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lv.wikipedia.org/wiki/Virsma" title="Virsma – Lets" lang="lv" hreflang="lv" data-title="Virsma" data-language-autonym="Latviešu" data-language-local-name="Lets" class="interlanguage-link-target"><span>Latviešu</span></a></li> <li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mr.wikipedia.org/wiki/%25E0%25A4%2586%25E0%25A4%25A1" title="आड – Marathi" lang="mr" hreflang="mr" data-title="आड" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nn.wikipedia.org/wiki/Flate" title="Flate – Noors - Nynorsk" lang="nn" hreflang="nn" data-title="Flate" data-language-autonym="Norsk nynorsk" data-language-local-name="Noors - Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li> <li class="interlanguage-link interwiki-no mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://no.wikipedia.org/wiki/Flate" title="Flate – Noors - Bokmål" lang="nb" hreflang="nb" data-title="Flate" data-language-autonym="Norsk bokmål" data-language-local-name="Noors - Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li> <li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oc.wikipedia.org/wiki/Superf%25C3%25ADcia_(matematicas)" title="Superfícia (matematicas) – Occitaans" lang="oc" hreflang="oc" data-title="Superfícia (matematicas)" data-language-autonym="Occitan" data-language-local-name="Occitaans" class="interlanguage-link-target"><span>Occitan</span></a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Powierzchnia" title="Powierzchnia – Pools" lang="pl" hreflang="pl" data-title="Powierzchnia" data-language-autonym="Polski" data-language-local-name="Pools" class="interlanguage-link-target"><span>Polski</span></a></li> <li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pms.wikipedia.org/wiki/Surfassa" title="Surfassa – Piëmontees" lang="pms" hreflang="pms" data-title="Surfassa" data-language-autonym="Piemontèis" data-language-local-name="Piëmontees" class="interlanguage-link-target"><span>Piemontèis</span></a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Superf%25C3%25ADcie" title="Superfície – Portugees" lang="pt" hreflang="pt" data-title="Superfície" data-language-autonym="Português" data-language-local-name="Portugees" class="interlanguage-link-target"><span>Português</span></a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Suprafa%25C8%259B%25C4%2583" title="Suprafață – Roemeens" lang="ro" hreflang="ro" data-title="Suprafață" data-language-autonym="Română" data-language-local-name="Roemeens" class="interlanguage-link-target"><span>Română</span></a></li> <li class="interlanguage-link interwiki-rsk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://rsk.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D0%25BE%25D1%2581%25D1%2586" title="Поверхносц – Pannonian Rusyn" lang="rsk" hreflang="rsk" data-title="Поверхносц" data-language-autonym="Руски" data-language-local-name="Pannonian Rusyn" class="interlanguage-link-target"><span>Руски</span></a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D0%25BE%25D1%2581%25D1%2582%25D1%258C" title="Поверхность – Russisch" lang="ru" hreflang="ru" data-title="Поверхность" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li> <li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sh.wikipedia.org/wiki/Povr%25C5%25A1" title="Površ – Servo-Kroatisch" lang="sh" hreflang="sh" data-title="Površ" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Servo-Kroatisch" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Surface" title="Surface – Simple English" lang="en-simple" hreflang="en-simple" data-title="Surface" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li> <li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Povrch" title="Povrch – Slowaaks" lang="sk" hreflang="sk" data-title="Povrch" data-language-autonym="Slovenčina" data-language-local-name="Slowaaks" class="interlanguage-link-target"><span>Slovenčina</span></a></li> <li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Ploskev" title="Ploskev – Sloveens" lang="sl" hreflang="sl" data-title="Ploskev" data-language-autonym="Slovenščina" data-language-local-name="Sloveens" class="interlanguage-link-target"><span>Slovenščina</span></a></li> <li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sn.wikipedia.org/wiki/Chiso_(Chiumbwa)" title="Chiso (Chiumbwa) – Shona" lang="sn" hreflang="sn" data-title="Chiso (Chiumbwa)" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li> <li class="interlanguage-link interwiki-so mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://so.wikipedia.org/wiki/Oogo_(dhul)" title="Oogo (dhul) – Somalisch" lang="so" hreflang="so" data-title="Oogo (dhul)" data-language-autonym="Soomaaliga" data-language-local-name="Somalisch" class="interlanguage-link-target"><span>Soomaaliga</span></a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D1%2580%25D1%2588" title="Површ – Servisch" lang="sr" hreflang="sr" data-title="Површ" data-language-autonym="Српски / srpski" data-language-local-name="Servisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Yta" title="Yta – Zweeds" lang="sv" hreflang="sv" data-title="Yta" data-language-autonym="Svenska" data-language-local-name="Zweeds" class="interlanguage-link-target"><span>Svenska</span></a></li> <li class="interlanguage-link interwiki-te mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://te.wikipedia.org/wiki/%25E0%25B0%2589%25E0%25B0%25AA%25E0%25B0%25B0%25E0%25B0%25BF%25E0%25B0%25A4%25E0%25B0%25B2%25E0%25B0%2582" title="ఉపరితలం – Telugu" lang="te" hreflang="te" data-title="ఉపరితలం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li> <li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/wiki/Y%25C3%25BCzey" title="Yüzey – Turks" lang="tr" hreflang="tr" data-title="Yüzey" data-language-autonym="Türkçe" data-language-local-name="Turks" class="interlanguage-link-target"><span>Türkçe</span></a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D1%258F" title="Поверхня – Oekraïens" lang="uk" hreflang="uk" data-title="Поверхня" data-language-autonym="Українська" data-language-local-name="Oekraïens" class="interlanguage-link-target"><span>Українська</span></a></li> <li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ur.wikipedia.org/wiki/%25D8%25B3%25D8%25B7%25D8%25AD" title="سطح – Urdu" lang="ur" hreflang="ur" data-title="سطح" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li> <li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uz.wikipedia.org/wiki/Sirt" title="Sirt – Oezbeeks" lang="uz" hreflang="uz" data-title="Sirt" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Oezbeeks" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li> <li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vec.wikipedia.org/wiki/Superficie" title="Superficie – Venetiaans" lang="vec" hreflang="vec" data-title="Superficie" data-language-autonym="Vèneto" data-language-local-name="Venetiaans" class="interlanguage-link-target"><span>Vèneto</span></a></li> <li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vi.wikipedia.org/wiki/M%25E1%25BA%25B7t_(t%25C3%25B4_p%25C3%25B4)" title="Mặt (tô pô) – Vietnamees" lang="vi" hreflang="vi" data-title="Mặt (tô pô)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamees" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> <li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://wuu.wikipedia.org/wiki/%25E6%259B%25B2%25E9%259D%25A2" title="曲面 – Wuyu" lang="wuu" hreflang="wuu" data-title="曲面" data-language-autonym="吴语" data-language-local-name="Wuyu" class="interlanguage-link-target"><span>吴语</span></a></li> <li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh.wikipedia.org/wiki/%25E6%259B%25B2%25E9%259D%25A2" title="曲面 – Chinees" lang="zh" hreflang="zh" data-title="曲面" data-language-autonym="中文" data-language-local-name="Chinees" class="interlanguage-link-target"><span>中文</span></a></li> <li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-yue.wikipedia.org/wiki/%25E6%259B%25B2%25E9%259D%25A2" title="曲面 – Kantonees" lang="yue" hreflang="yue" data-title="曲面" data-language-autonym="粵語" data-language-local-name="Kantonees" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 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