Surface (topology)

<p>In the part of mathematics referred to as <a href="/facts/Topology/2EqW47cU">topology</a>, a surface is a two-dimensional <a href="/facts/Manifold/rXPERJtu">manifold</a>. Some surfaces arise as the <a href="/facts/Manifold/rXPERJtu">boundaries</a> of <a href="/facts/Three-dimensional/KAqKWUbS">three-dimensional</a> <a href="/facts/Solid_figure/tHH3JFoq">solid figures</a>; for example, the sphere is the boundary of the solid <a href="/facts/Ball_(mathematics)/f9vpMVMp">ball</a>. Other surfaces arise as graphs of <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any <a href="/facts/Ambient_space/umgCRklf">ambient space</a>. For example, the <a href="/facts/Klein_bottle/LTenG5E2">Klein bottle</a> is a surface that cannot be <a href="/facts/Embedding/HKeVKc42">embedded</a> in three-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>.
</p><p>Topological surfaces are sometimes equipped with additional information, such as a <a href="/facts/Riemannian_metric/5KYnmEgF">Riemannian metric</a> or a complex structure, that connects them to other disciplines within mathematics, such as <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a> and <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>. The various <a href="/facts/Surface_(mathematics)/whF08jvy">mathematical notions of surface</a> can be used to model <a href="/facts/Surface/0XA8ce89">surfaces</a> in the physical world.
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Surface (topology) open-in-new

Surface (topology)