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class="box-Kaynaksız plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"> <tbody> <tr> <td class="mbox-text"> <div class="mbox-text-span"> Bu madde <b>hiçbir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Vikipedi:Kaynak_g%C3%B6sterme?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vikipedi:Kaynak gösterme">kaynak</a> <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Vikipedi:Do%C4%9Frulanabilirlik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vikipedi:Doğrulanabilirlik">içermemektedir</a>.</b><span class="hide-when-compact"> Lütfen <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Yard%C4%B1m:Dipnotlar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Yardım:Dipnotlar">güvenilir kaynaklar ekleyerek</a> <a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/w/index.php?title%3DY%25C3%25BCzey%26action%3Dedit">madde içeriğinin geliştirilmesine</a> yardımcı olun. Kaynaksız içerik itiraz konusu olabilir ve <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Vikipedi:Do%C4%9Frulanabilirlik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Kan%C4%B1t_sorumlulu%C4%9Fu" title="Vikipedi:Doğrulanabilirlik">kaldırılabilir</a>.<br><small><span class="plainlinks"><i>Kaynak ara:</i> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.google.com/search?as_eq%3Dwikipedia%26q%3D%2522Y%25C3%25BCzey%2522">"Yüzey"</a> – <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.google.com/search?tbm%3Dnws%26q%3D%2522Y%25C3%25BCzey%2522%2B-wikipedia">haber</a> · <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.google.com/search?%26q%3D%2522Y%25C3%25BCzey%2522%2Bsite:news.google.com/newspapers%26source%3Dnewspapers">gazete</a> · <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.google.com/search?tbs%3Dbks:1%26q%3D%2522Y%25C3%25BCzey%2522%2B-wikipedia">kitap</a> · <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://scholar.google.com/scholar?q%3D%2522Y%25C3%25BCzey%2522">akademik</a> · <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.jstor.org/action/doBasicSearch?Query%3D%2522Y%25C3%25BCzey%2522%26acc%3Don%26wc%3Don">JSTOR</a></span></small></span> <small class="date-container"><i>(<span class="date">Ocak 2012</span>)</i></small><small class="hide-when-compact"><i> (<a href="https://tr-m-wikipedia-org.translate.goog/wiki/Yard%C4%B1m:Bak%C4%B1m_%C5%9Fablonunu_kald%C4%B1rmak?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Yardım:Bakım şablonunu kaldırmak">Bu şablonun nasıl ve ne zaman kaldırılması gerektiğini öğrenin</a>)</i></small> </div></td> </tr> </tbody> </table> <p><b>Yüzey</b>, <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Matematik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matematik">matematikte</a> ve özellikle <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Topoloji?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Topoloji">topolojide</a> iki <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Boyut?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Boyut">boyutlu</a> <a href="https://tr-m-wikipedia-org.translate.goog/wiki/%C3%87okkatl%C4%B1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Çokkatlı">çokkatlı</a>. İki <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Ger%C3%A7el_say%C4%B1lar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Gerçel sayılar">gerçel</a> değişkenli ve gerçel değerli bir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Fonksiyon?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fonksiyon">fonksiyonun</a> <a href="https://tr-m-wikipedia-org.translate.goog/wiki/%C3%9C%C3%A7_boyut?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Üç boyut">üç boyutlu</a> <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Uzay?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uzay">uzayda</a> (<b>R</b>³) grafiği tipik yüzey örneğidir. Ayrıca Dünya yüzeyi, bir yumurtanın kabuğu, bir <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Simit_(Topoloji)&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Simit (Topoloji) (sayfa mevcut değil)">simit</a> birer yüzeydir.</p> <p>Bir yüzeyin iki boyutlu bir çokkatlı olması, öncelikle onun (belirli özellikleri sağlayan) bir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Topolojik_uzaylar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Topolojik uzaylar">topolojik uzay</a> olması demektir. Bunun yanında yüzeyin verilen (herhangi) bir <i>x</i> noktası çevresinde öyle bir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Kom%C5%9Fuluk_(matematik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Komşuluk (matematik)">komşuluk</a> bulunabilir ki, bu komşuluk 2 boyutlu uzayın bir parçasına <i>benzer</i>. Bu komşuluğa <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Yama_(Topoloji)&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Yama (Topoloji) (sayfa mevcut değil)">yama</a> denir. Bu benzeme uyarınca, <i>x</i> çevresinde sağ-sol ve yukarı-aşağı kavramları iyi bir biçimde tanımlanabilir. Daha iyi bir deyişle, <i>x'</i>in çevresine bir koordinat sistemi döşenebilir. Böylece yüzey, bir düzlem parçası olmasa bile <i>x</i> çevresindeki noktalar bir düzlemdeymiş gibi koordinatlara sahip olur.</p> <p>Dünya yüzeyi matematiksel olarak bir yüzeydir. Dünya'nın çizilen her haritası, yukarıdaki anlamda bir koordinat sistemi tarif eder. Bu sayede denizcilikte yön bulma kolaylaşır ve iki denizci aynı koordinat sisteminde konuşarak birbirleriyle anlaşabilir. Dünya yüzeyi için standart koordinat sistemi, enlem ve boylamlarla verilir. Örneğin, Dünya yüzeyinden gün dönümü çizgisi ve kutuplar silindiğinde kalan parçaya (<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 180^{\circ }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn> 180 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> ∘ </mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 180^{\circ }} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d0431ce231935522dc0cb52df7f2b406cdadc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 180^{\circ }}"></span> <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Do%C4%9Fu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Doğu">Doğu</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 180^{\circ }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn> 180 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> ∘ </mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 180^{\circ }} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d0431ce231935522dc0cb52df7f2b406cdadc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 180^{\circ }}"></span> <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Bat%C4%B1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Batı">Batı</a>) ilâ (<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 90^{\circ }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn> 90 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> ∘ </mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 90^{\circ }} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}"></span> <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Kuzey?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kuzey">Kuzey</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 90^{\circ }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn> 90 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> ∘ </mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 90^{\circ }} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}"></span> <a href="https://tr-m-wikipedia-org.translate.goog/wiki/G%C3%BCney?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Güney">Güney</a>) koordinatları verilerek bu parça bir yamaya dönüştürülebilir. Gün dönümü çizgisi ya da kutupların silinmediği durumda bazı enlem-boylam çiftlerinin aynı noktayı tarif edeceklerine dikkat ediniz.</p> <p>Topolojik bir yüzey, her zaman <b>R</b>³'te görülemeyebilir. Örneğin <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Ger%C3%A7el_izd%C3%BC%C5%9F%C3%BCmsel_d%C3%BCzlem&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Gerçel izdüşümsel düzlem (sayfa mevcut değil)">gerçel izdüşümsel düzlem</a> ya da <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Klein_%C5%9Fi%C5%9Fesi&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Klein şişesi (sayfa mevcut değil)">Klein şişesi</a> <b>R</b>³'te <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=G%C3%B6mme_(Topoloji)&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Gömme (Topoloji) (sayfa mevcut değil)">yatmazlar</a> ancak <b>R</b><sup>4</sup>'e <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=G%C3%B6mme_(Topoloji)&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Gömme (Topoloji) (sayfa mevcut değil)">gömülebilirler</a>. Topolojinin temel teoremlerinden biri, bir yüzeyi gömebilmek için en fazla dört boyuta (<b>R</b><sup>4</sup>) gerek olduğunu söyler.</p> </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="Matematiksel_tanım"><span id="Matematiksel_tan.C4.B1m"></span>Matematiksel tanım</h2><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Y%C3%BCzey&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Matematiksel tanım" 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>değiştir</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>İki boyutlu bir çokkatlıya <b>yüzey</b> denir. Daha ayrıntılı bir söyleyişle, (<b><a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Kenar_(Topoloji)&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Kenar (Topoloji) (sayfa mevcut değil)">kenarı</a> olmayan topolojik</b>) <b>yüzey</b>, aşağıdaki koşulları sağlayan bir topolojik uzaydır:</p> <ul> <li><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Hausdorff_uzay?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hausdorff uzay">Hausdorff</a>'tur;</li> <li>Herhangi bir noktasının çevresinde öyle bir <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=A%C3%A7%C4%B1k_K%C3%BCme&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Açık Küme (sayfa mevcut değil)">açık</a> <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Kom%C5%9Fuluk_(Topoloji)&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Komşuluk (Topoloji) (sayfa mevcut değil)">komşuluk</a> bulunabilir ki bu komşuluk <b>R</b>²'nin açık bir alt kümesine <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Homeomorfizma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Homeomorfizma">homeomorfiktir</a>;</li> <li>(Kimi tanımlarda) <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=%C4%B0kinci_say%C4%B1labilirlik&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="İkinci sayılabilirlik (sayfa mevcut değil)">İkinci sayılabilirlik</a> özelliğini sağlar;</li> <li>(Kimi tanımlarda) <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Parakompakt&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Parakompakt (sayfa mevcut değil)">Parakompakttır</a>.</li> </ul> <p>Yukarıki tanımda ikinci koşulda <b>R</b>² yerine, üst yarı düzlemi (yani ikinci koordinatları negatif olmayan noktaların kümesi) temsil etmek üzere <b>H</b>² konduğunda, bu tanım, <b>kenarı olan (kenarlı) topolojik bir yüzey</b> tanımına dönüşür. Bu durumda ikinci koşulda homeomorfizma sözcüğünün anlamlı olabilmesi için <b>H</b>² üzerinde bir topoloji bulunması gerekir. Bu topoloji standart olarak <b>R</b>²'den <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Alt_uzay_topolojisi&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Alt uzay topolojisi (sayfa mevcut değil)">tetiklenen topolojidir</a>.</p> <p>Kenarı olan bir yüzeyin kenarı olmayandan farklı olarak şu tür noktaları da vardır: noktanın yeterince küçük her komşuluğu <b>H</b>²'de çapı yarı düzlemin en altında oturan bir yarım daireye homeomorfiktir. Noktanın <b>R</b>²'de açık bir bölgeye homeomorfik bir komşuluğu olması söz konusu değildir. Kenarlı yüzeylere birkaç örnek: düzlemde <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Kapal%C4%B1_K%C3%BCme&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Kapalı Küme (sayfa mevcut değil)">kapalı</a> bir daire, kapalı bir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/E%C4%9Fri?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eğri">eğriyle</a> çevrelenmiş bir düzlem bölgesi, bir yarıküre (içi boş), açık bir dairesel parçası koparılmış bir simit (yüzeyi).</p> <p>Bir yüzeyin içinde bir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/M%C3%B6bius_%C5%9Feridi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Möbius şeridi">Möbius şeridi</a> varsa (yüzeye gömülebiliyorsa) bu yüzeye <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Y%C3%B6n_Verilebilirlik&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Yön Verilebilirlik (sayfa mevcut değil)"><b>yön verilemez</b></a> denir. İçinde bir Möbius şeridi yoksa böyle bir yüzeye <b>yön verilebilir</b> denir. Yön verilemez yüzeylere birkaç örnek: Möbius şeridi, gerçel izdüşümsel düzlem, Klein şişesi. Bunlardan Möbius şeridi kenarı (bir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/%C3%87ember?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Çember">çember</a>) olan bir yüzeyken diğerleri kenarsız yüzeylerdir.</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="Yüzeylerin_sınıflandırılması"><span id="Y.C3.BCzeylerin_s.C4.B1n.C4.B1fland.C4.B1r.C4.B1lmas.C4.B1"></span>Yüzeylerin sınıflandırılması</h2><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Y%C3%BCzey&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Yüzeylerin sınıflandırılması" 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>değiştir</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>Matematiğin temel uğraşlarından biri sınıflamadır. Tanımladığı bir nesne türünde, nesnelerin bazılarını bibirinden ayırt etmeden, olası tüm nesneleri listelemek sınıflamadaki amaçtır. Dolayısıyla yukarıda soyut tanımı verilen yüzeylerin tümünü listelemek, topolojinin ilgilendiği bir sorudur. Bunu yaparken iki homeomorfik yüzeyi bir tutar, bunların arasında ayrım gözetmez. Bu koşullar altında listeyi oluşturmaya çalışır. Örneğin bu listede (içi boş) bir küp ve bir <a href="https://tr-m-wikipedia-org.translate.goog/wiki/K%C3%BCre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Küre">küre</a> birlikte görünmeyecektir; yalnızca biri listede yer alacaktır çünkü bu iki yüzey, <b>R</b>³'ten tetiklenen topolojileriyle birbirine <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Homeomorfizma?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Homeomorfizma">homeomorfik</a> yüzeylerdir.</p> <p>Şu ve benzeri soruların yanıtlanması gerekir: Bir küreyle bir simit, Möbius şeridiyle daire, kenarı olan yüzeyle olmayan ve yön verilebilir olanla olmayan birbirine homeomorfik midir?</p> <p>Yüzeylerin sınıflandırılması problemi ilk kez <a href="https://tr-m-wikipedia-org.translate.goog/wiki/August_Ferdinand_M%C3%B6bius?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="August Ferdinand Möbius">August Ferdinand Möbius</a> tarafından çalışılmış ve <b>R</b>³'te yatan yön verilebilir yüzeyler için 1870 yılında sonuç ilan edilmiştir. Max Wilhelm Dehn ve P. Heegard 1907 yılında <a href="https://tr-m-wikipedia-org.translate.goog/wiki/%C3%9C%C3%A7genlenebilir?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Üçgenlenebilir">üçgenlenebilir</a> yüzeyler için tüm sınıflandırmayı vermiştir. Her topolojik yüzeyin üçgenlenebilir olduğunu 1925 yılında Tibor Radó ispatlayarak sınıflandırmayı sona erdirmiştir (ispat için L. V. Ahlfors ve L. Sario'nun aşağıda listelenmiş kitabına bakınız).</p> <p>Bu sınıflandırmaya göre, <i><a href="https://tr-m-wikipedia-org.translate.goog/wiki/T%C4%B1k%C4%B1zl%C4%B1k?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tıkızlık">tıkız</a>, yön verilebilir, kenarsız</i> yüzeyler, şunlardan biri(ne homeomorfik) olmak zorundadır:</p> <p><span class="mw-default-size" typeof="mw:File"><a href="https://tr-m-wikipedia-org.translate.goog/wiki/Dosya:Y%C3%BCzey_s%C4%B1n%C4%B1fland%C4%B1rmas%C4%B1.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/tr/6/64/Y%C3%BCzey_s%C4%B1n%C4%B1fland%C4%B1rmas%C4%B1.jpg" decoding="async" width="674" height="148" class="mw-file-element" data-file-width="674" data-file-height="148"> </noscript><span class="lazy-image-placeholder" style="width: 674px;height: 148px;" data-src="//upload.wikimedia.org/wikipedia/tr/6/64/Y%C3%BCzey_s%C4%B1n%C4%B1fland%C4%B1rmas%C4%B1.jpg" data-width="674" data-height="148" data-class="mw-file-element"> </span></a></span></p> <p>İlk şekil bir küredir (<b>S</b>²). İkincisi bir simit (<b>T</b>²). Üçüncü şekil çift <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Delik_(Topoloji)&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Delik (Topoloji) (sayfa mevcut değil)">delikli</a> bir yüzeyi (<b>F</b>²) anlatır. Listede sırasıyla 3, 4, 5, … delikli yüzeyler (sırasıyla <b>F</b><sub>3</sub>, <b>F</b><sub>4</sub>, <b>F</b><sub>5</sub>, …) yer alacaktır. Dikkat edilirse, iki ayrı simitten birer daire oyulup kalan yüzeyler birbirlerine yapıştırılırsa, çıkan yüzey, iki delikli bir yüzey olacaktır. Üzerindeki topoloji, bu yapıştırma sırasında kullanılan özdeşleştirme aracılığıyla gelen <a href="https://tr-m-wikipedia-org.translate.goog/wiki/B%C3%B6l%C3%BCm_topolojisi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bölüm topolojisi">bölüm topolojisidir</a>. Bu işlem şöyle gösterilir:</p> <p><b>F</b><sub>2</sub> = <b>T</b>² # <b>T</b>²</p> <p>Daireler oyarak yapıştırma işlemine <a href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Ba%C4%9Flant%C4%B1l%C4%B1_toplam&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Bağlantılı toplam (sayfa mevcut değil)">bağlantılı toplam</a> denir. Üç delikli bir yüzey, çift delikli bir yüzeyle torusun bağlantılı toplamı olarak inşa edilebilir.</p> <p>Bu sınıflandırmadan anlaşılıyor ki, tıkız, yön verilebilir, kenarsız yüzeyler delik sayılarıyla anlatılabilirler. Kürenin delik sayısına 0 diyoruz. Simidin delik sayısı 1'dir.</p> <p>Tıkız, yön verilebilir, kenarsız <b>S</b> adlı bir yüzey için 2 - 2<i>g</i> sayısı yüzeyin <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Euler_say%C4%B1s%C4%B1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Euler sayısı">Euler sayısına</a> eşittir ve şöyle gösterilir:</p> <p><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 (S)=2-2g}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> χ </mi> <mo stretchy="false"> ( </mo> <mi> S </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 2 </mn> <mo> − </mo> <mn> 2 </mn> <mi> g </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \chi (S)=2-2g} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21de672b1953817ed423e8f4c008498a81341292" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.143ex; height:2.843ex;" alt="{\displaystyle \chi (S)=2-2g}"> </noscript><span class="lazy-image-placeholder" style="width: 14.143ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21de672b1953817ed423e8f4c008498a81341292" data-alt="{\displaystyle \chi (S)=2-2g}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <p>Yön verilemez yüzeyler için sınıflandırmaysa temelde aynı olmasına karşın, söz konusu yüzeylere daha az aşinayız. Tıkız, <i>yön verilemez</i>, kenarsız yüzeylerin en <i>basiti</i> <b>gerçel izdüşümsel düzlemdir</b> (<b>RP</b>²). Bu yüzey, bir Möbius şeridiyle bir dairenin kenarlarından birbirlerine yapıştırılmasıyla inşa edilir. Üzerindeki topoloji, bu yapıştırma aracılığıyla gelen bölüm topolojisidir. İki tane <b>RP</b>²'nin bağlantılı toplamına <b>Klein şişesi</b> (<b>K</b>²) denir:</p> <p><b>K</b>² = <b>RP</b>² # <b>RP</b>²</p> <p>Bu işlem iki Möbius şeridinin kenarlarından birbirlerine yapıştırılmasından başka bir şey değildir.</p> <p>Sınıflandırma şunu söyler: <i>tıkız, yön verilemez, kenarsız</i> yüzeyler aşağıdakilerden biri(ne homeomorfik) olmak zorundadır:</p> <p><b>RP</b>², <b>K</b>² = <b>RP</b>² # <b>RP</b>², (<b>RP</b>² # <b>RP</b>² # <b>RP</b>²), (<b>RP</b>² # <b>RP</b>² # <b>RP</b>² # <b>RP</b>²), <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 \cdots }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> ⋯ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \cdots } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d67495288eac0fa90d5bbcad7d9a343c15ad56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:2.723ex; height:1.176ex;" alt="{\displaystyle \cdots }"> </noscript><span class="lazy-image-placeholder" style="width: 2.723ex;height: 1.176ex;vertical-align: 0.439ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d67495288eac0fa90d5bbcad7d9a343c15ad56" data-alt="{\displaystyle \cdots }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>Gösterilebilir ki bu listedeki yüzeylerin Euler sayıları 1'den başlar ve birer birer azalır:</p> <p><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 (RP^{2})=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> χ </mi> <mo stretchy="false"> ( </mo> <mi> R </mi> <msup> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \chi (RP^{2})=1} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b5c0a32fdb96d39421ef994f9e91abb08adcd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.165ex; height:3.176ex;" alt="{\displaystyle \chi (RP^{2})=1}"> </noscript><span class="lazy-image-placeholder" style="width: 12.165ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b5c0a32fdb96d39421ef994f9e91abb08adcd9" data-alt="{\displaystyle \chi (RP^{2})=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p><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 (RP^{2}\#RP^{2})=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> χ </mi> <mo stretchy="false"> ( </mo> <mi> R </mi> <msup> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi mathvariant="normal"> # </mi> <mi> R </mi> <msup> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \chi (RP^{2}\#RP^{2})=0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504dc030b18a6fcb9575b9c70b2d9314e86ece5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.741ex; height:3.176ex;" alt="{\displaystyle \chi (RP^{2}\#RP^{2})=0}"> </noscript><span class="lazy-image-placeholder" style="width: 18.741ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504dc030b18a6fcb9575b9c70b2d9314e86ece5e" data-alt="{\displaystyle \chi (RP^{2}\#RP^{2})=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p><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 (RP^{2}\#RP^{2}\#RP^{2})=-1...}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> χ </mi> <mo stretchy="false"> ( </mo> <mi> R </mi> <msup> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi mathvariant="normal"> # </mi> <mi> R </mi> <msup> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi mathvariant="normal"> # </mi> <mi> R </mi> <msup> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo> − </mo> <mn> 1... </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \chi (RP^{2}\#RP^{2}\#RP^{2})=-1...} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c8d01d50db32263bb8908bb42e4afdeaa41f56a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.065ex; height:3.176ex;" alt="{\displaystyle \chi (RP^{2}\#RP^{2}\#RP^{2})=-1...}"> </noscript><span class="lazy-image-placeholder" style="width: 29.065ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c8d01d50db32263bb8908bb42e4afdeaa41f56a" data-alt="{\displaystyle \chi (RP^{2}\#RP^{2}\#RP^{2})=-1...}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>Dolayısıyla, (tıkız, kenarsız) bir yüzeyin <a href="https://tr-m-wikipedia-org.translate.goog/wiki/Euler_say%C4%B1s%C4%B1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Euler sayısı">Euler sayısını</a> ve yön verilebilir olup olmadığını söylemek, yüzeyi anlatmaya yeter.</p> <div class="mw-heading mw-heading3"> <h3 id="Kaynakça"><span id="Kaynak.C3.A7a"></span>Kaynakça</h3><span class="mw-editsection"> <a role="button" href="https://tr-m-wikipedia-org.translate.goog/w/index.php?title=Y%C3%BCzey&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Değiştirilen bölüm: Kaynakça" 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>değiştir</span> </a> </span> </div> <style data-mw-deduplicate="TemplateStyles:r32805677">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-count:2}.mw-parser-output .reflist-columns-3{column-count:3}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style> <div class="reflist"> </div> <p><small> </small></p><small> <ul> <li><cite class="kaynak kitap">Hatcher, Allen (2002). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/algebraictopolog0000hatc"><i>Algebraic topology</i></a>. Cambridge: Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+topology&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2002&rft.aulast=Hatcher&rft.aufirst=Allen&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebraictopolog0000hatc&rfr_id=info%3Asid%2Ftr.wikipedia.org%3AY%C3%BCzey" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="kaynak kitap">Munkres, James R. (2000). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/topology00munk"><i>Topology (Second Edition)</i></a>. Prentice Hall. s. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/topology00munk/page/n546">537</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology+%28Second+Edition%29&rft.pages=537&rft.pub=Prentice+Hall&rft.date=2000&rft.aulast=Munkres&rft.aufirst=James+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftopology00munk&rfr_id=info%3Asid%2Ftr.wikipedia.org%3AY%C3%BCzey" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="kaynak kitap">Ahlfors, Lars V.; Sario, Leo (1960). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/riemannsurfaces0000ahlf"><i>Riemann surfaces</i></a>. Princeton: Princeton University Press. s. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/riemannsurfaces0000ahlf/page/382">382</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Riemann+surfaces&rft.place=Princeton&rft.pages=382&rft.pub=Princeton+University+Press&rft.date=1960&rft.aulast=Ahlfors&rft.aufirst=Lars+V.&rft.au=Sario%2C+Leo&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Friemannsurfaces0000ahlf&rfr_id=info%3Asid%2Ftr.wikipedia.org%3AY%C3%BCzey" class="Z3988"><span style="display:none;"> </span></span></li> </ul></small><small></small> </section> </div> <noscript> <img src="https://login.m.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&mobile=1" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> "<a dir="ltr" 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href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ast.wikipedia.org/wiki/Superficie" title="Superficie - Asturyasça" lang="ast" hreflang="ast" data-title="Superficie" data-language-autonym="Asturianu" data-language-local-name="Asturyasça" 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 - Azerbaycan dili" lang="az" hreflang="az" data-title="Səth" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaycan dili" 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 - Belarusça" lang="be" hreflang="be" data-title="Паверхня" data-language-autonym="Беларуская" data-language-local-name="Belarusça" 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 - Bulgarca" lang="bg" hreflang="bg" data-title="Повърхност" data-language-autonym="Български" data-language-local-name="Bulgarca" 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) - Bengalce" lang="bn" hreflang="bn" data-title="তল (টপোলজি)" data-language-autonym="বাংলা" data-language-local-name="Bengalce" 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 - Boşnakça" lang="bs" hreflang="bs" data-title="Površ" data-language-autonym="Bosanski" data-language-local-name="Boşnakça" 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) - Katalanca" lang="ca" hreflang="ca" data-title="Superfície (matemàtiques)" data-language-autonym="Català" data-language-local-name="Katalanca" 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 - Orta Kürtçe" lang="ckb" hreflang="ckb" data-title="ڕوو" data-language-autonym="کوردی" data-language-local-name="Orta Kürtçe" 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 - Çekçe" lang="cs" hreflang="cs" data-title="Plocha" data-language-autonym="Čeština" data-language-local-name="Çekçe" 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 - Çuvaşça" lang="cv" hreflang="cv" data-title="Çий" data-language-autonym="Чӑвашла" data-language-local-name="Çuvaşça" 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) - İngilizce" lang="en" hreflang="en" data-title="Surface (topology)" data-language-autonym="English" data-language-local-name="İngilizce" 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) - İspanyolca" lang="es" hreflang="es" data-title="Superficie (topología)" data-language-autonym="Español" data-language-local-name="İspanyolca" 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 - Estonca" lang="et" hreflang="et" data-title="Pind" data-language-autonym="Eesti" data-language-local-name="Estonca" 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 - Baskça" lang="eu" hreflang="eu" data-title="Gainazal" data-language-autonym="Euskara" data-language-local-name="Baskça" 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) - Farsça" lang="fa" hreflang="fa" data-title="رویه (توپولوژی)" data-language-autonym="فارسی" data-language-local-name="Farsça" 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) - Fince" lang="fi" hreflang="fi" data-title="Pinta (geometria)" data-language-autonym="Suomi" data-language-local-name="Fince" 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) - Fransızca" lang="fr" hreflang="fr" data-title="Surface (géométrie)" data-language-autonym="Français" data-language-local-name="Fransızca" 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 - Friuli dili" lang="fur" hreflang="fur" data-title="Superficie" data-language-autonym="Furlan" data-language-local-name="Friuli dili" 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) - İrlandaca" lang="ga" hreflang="ga" data-title="Dromchla (toipeolaíocht)" data-language-autonym="Gaeilge" data-language-local-name="İrlandaca" 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 - Galiçyaca" lang="gl" hreflang="gl" data-title="Superficie" data-language-autonym="Galego" data-language-local-name="Galiçyaca" 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) - İbranice" lang="he" hreflang="he" data-title="משטח (טופולוגיה)" data-language-autonym="עברית" data-language-local-name="İbranice" 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 - Hintçe" lang="hi" hreflang="hi" data-title="पृष्ट" data-language-autonym="हिन्दी" data-language-local-name="Hintçe" 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) - Hırvatça" lang="hr" hreflang="hr" data-title="Ploha (geometrija)" data-language-autonym="Hrvatski" data-language-local-name="Hırvatça" 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 - Macarca" lang="hu" hreflang="hu" data-title="Felszín" data-language-autonym="Magyar" data-language-local-name="Macarca" 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 - Ermenice" lang="hy" hreflang="hy" data-title="Մակերևույթ" data-language-autonym="Հայերեն" data-language-local-name="Ermenice" 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 - İnterlingua" lang="ia" hreflang="ia" data-title="Superficie" data-language-autonym="İnterlingua" data-language-local-name="İnterlingua" class="interlanguage-link-target"><span>İnterlingua</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) - Endonezce" lang="id" hreflang="id" data-title="Permukaan (topologi)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Endonezce" 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 - İnguşça" lang="inh" hreflang="inh" data-title="ТӀехе" data-language-autonym="ГӀалгӀай" data-language-local-name="İnguşça" 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 - İzlandaca" lang="is" hreflang="is" data-title="Yfirborð" data-language-autonym="Íslenska" data-language-local-name="İzlandaca" 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 - İtalyanca" lang="it" hreflang="it" data-title="Superficie" data-language-autonym="İtaliano" data-language-local-name="İtalyanca" class="interlanguage-link-target"><span>İtaliano</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 - Japonca" lang="ja" hreflang="ja" data-title="曲面" data-language-autonym="日本語" data-language-local-name="Japonca" 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) - Kazakça" lang="kk" hreflang="kk" data-title="Бет (геометрия)" data-language-autonym="Қазақша" data-language-local-name="Kazakça" 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 - Korece" lang="ko" hreflang="ko" data-title="곡면" data-language-autonym="한국어" data-language-local-name="Korece" 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) - Kırgızca" lang="ky" hreflang="ky" data-title="Бет (Геометрия)" data-language-autonym="Кыргызча" data-language-local-name="Kırgızca" 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) - Ligurca" lang="lij" hreflang="lij" data-title="Superfiçie (matematica)" data-language-autonym="Ligure" data-language-local-name="Ligurca" 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 - Litvanca" lang="lt" hreflang="lt" data-title="Paviršius" data-language-autonym="Lietuvių" data-language-local-name="Litvanca" 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 - Letonca" lang="lv" hreflang="lv" data-title="Virsma" data-language-autonym="Latviešu" data-language-local-name="Letonca" 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 dili" lang="mr" hreflang="mr" data-title="आड" data-language-autonym="मराठी" data-language-local-name="Marathi dili" class="interlanguage-link-target"><span>मराठी</span></a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Oppervlak_(topologie)" title="Oppervlak (topologie) - Felemenkçe" lang="nl" hreflang="nl" data-title="Oppervlak (topologie)" data-language-autonym="Nederlands" data-language-local-name="Felemenkçe" class="interlanguage-link-target"><span>Nederlands</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 - Norveççe Nynorsk" lang="nn" hreflang="nn" data-title="Flate" data-language-autonym="Norsk nynorsk" data-language-local-name="Norveççe 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 - Norveççe Bokmål" lang="nb" hreflang="nb" data-title="Flate" data-language-autonym="Norsk bokmål" data-language-local-name="Norveççe 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) - Oksitan dili" lang="oc" hreflang="oc" data-title="Superfícia (matematicas)" data-language-autonym="Occitan" data-language-local-name="Oksitan dili" 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 - Lehçe" lang="pl" hreflang="pl" data-title="Powierzchnia" data-language-autonym="Polski" data-language-local-name="Lehçe" 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 - Piyemontece" lang="pms" hreflang="pms" data-title="Surfassa" data-language-autonym="Piemontèis" data-language-local-name="Piyemontece" 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 - Portekizce" lang="pt" hreflang="pt" data-title="Superfície" data-language-autonym="Português" data-language-local-name="Portekizce" 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 - Rumence" lang="ro" hreflang="ro" data-title="Suprafață" data-language-autonym="Română" data-language-local-name="Rumence" 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 - Rusça" lang="ru" hreflang="ru" data-title="Поверхность" data-language-autonym="Русский" data-language-local-name="Rusça" 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 - Sırp-Hırvat Dili" lang="sh" hreflang="sh" data-title="Površ" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Sırp-Hırvat Dili" 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 - Slovakça" lang="sk" hreflang="sk" data-title="Povrch" data-language-autonym="Slovenčina" data-language-local-name="Slovakça" 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 - Slovence" lang="sl" hreflang="sl" data-title="Ploskev" data-language-autonym="Slovenščina" data-language-local-name="Slovence" 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) - Şona dili" lang="sn" hreflang="sn" data-title="Chiso (Chiumbwa)" data-language-autonym="ChiShona" data-language-local-name="Şona dili" 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) - Somalice" lang="so" hreflang="so" data-title="Oogo (dhul)" data-language-autonym="Soomaaliga" data-language-local-name="Somalice" 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 - Sırpça" lang="sr" hreflang="sr" data-title="Површ" data-language-autonym="Српски / srpski" data-language-local-name="Sırpça" 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 - İsveççe" lang="sv" hreflang="sv" data-title="Yta" data-language-autonym="Svenska" data-language-local-name="İsveççe" 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 dili" lang="te" hreflang="te" data-title="ఉపరితలం" data-language-autonym="తెలుగు" data-language-local-name="Telugu dili" class="interlanguage-link-target"><span>తెలుగు</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 - Ukraynaca" lang="uk" hreflang="uk" data-title="Поверхня" data-language-autonym="Українська" data-language-local-name="Ukraynaca" 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 - Urduca" lang="ur" hreflang="ur" data-title="سطح" data-language-autonym="اردو" data-language-local-name="Urduca" 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 - Özbekçe" lang="uz" hreflang="uz" data-title="Sirt" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Özbekçe" 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 - Venedikçe" lang="vec" hreflang="vec" data-title="Superficie" data-language-autonym="Vèneto" data-language-local-name="Venedikçe" 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) - Vietnamca" lang="vi" hreflang="vi" data-title="Mặt (tô pô)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamca" 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 - Wu Çincesi" lang="wuu" hreflang="wuu" data-title="曲面" data-language-autonym="吴语" data-language-local-name="Wu Çincesi" 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 - Çince" lang="zh" hreflang="zh" data-title="曲面" data-language-autonym="中文" data-language-local-name="Çince" 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 - Kantonca" lang="yue" hreflang="yue" data-title="曲面" data-language-autonym="粵語" data-language-local-name="Kantonca" 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-tr.svg" alt="Vikipedi" width="107" height="18" style="width: 6.6875em; height: 1.125em;"> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod">Sayfa en son 12.08, 21 Mayıs 2024 tarihinde değiştirildi.</li> <li id="footer-info-copyright">Aksi belirtilmedikçe içeriğin kullanımı <a class="external" rel="nofollow" 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