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In mathematics, specifically, in topology, a surface is a two-dimensional, topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3, such as a sphere. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections. To say that a surface is "two-dimensional" means that, about each point, there is a ''coordinate patch'' on which a two-dimensional coordinate system is defined. For example, the surface of the Earth is (ideally) a two-dimensional sphere, and latitude and longitude provide two-dimensional coordinates on it (except at the poles and along the 180th meridian). The concept of surface finds application in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface. ==Definitions and first examples== A ''(topological) surface'' is a topological space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E2. Such a neighborhood, together with the corresponding homeomorphism, is known as a ''(coordinate) chart''. It is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. These coordinates are known as ''local coordinates'' and these homeomorphisms lead us to describe surfaces as being ''locally Euclidean''. In most writings on the subject, it is often assumed, explicitly or implicitly, that as a topological space a surface is also nonempty, second countable, and Hausdorff. It is also often assumed that the surfaces under consideration are connected. The rest of this article will assume, unless specified otherwise, that a surface is nonempty, Hausdorff, second countable, and connected. More generally, a ''(topological) surface with boundary'' is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the upper half-plane H2 in C. These homeomorphisms are also known as ''(coordinate) charts''. The boundary of the upper half-plane is the ''x''-axis. A point on the surface mapped via a chart to the ''x''-axis is termed a ''boundary point''. The collection of such points is known as the ''boundary'' of the surface which is necessarily a one-manifold, that is, the union of closed curves. On the other hand, a point mapped to above the ''x''-axis is an ''interior point''. The collection of interior points is the ''interior'' of the surface which is always non-empty. The closed disk is a simple example of a surface with boundary. The boundary of the disc is a circle. The term ''surface'' used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary which is compact is known as a 'closed' surface. The two-dimensional sphere, the two-dimensional torus, and the real projective plane are examples of closed surfaces. The Möbius strip is a surface on which the distinction between clockwise and counterclockwise can be defined locally, but not globally. In general, a surface is said to be ''orientable'' if it does not contain a homeomorphic copy of the Möbius strip; intuitively, it has two distinct "sides". For example, the sphere and torus are orientable, while the real projective plane is not (because the real projective plane with one point removed is homeomorphic to the open Möbius strip). In differential and algebraic geometry, extra structure is added upon the topology of the surface. This added structures can be a smoothness structure (making it possible to define differentiable maps to and from the surface), a Riemannian metric (making it possible to define length and angles on the surface), a complex structure (making it possible to define holomorphic maps to and from the surface — in which case the surface is called a Riemann surface), or an algebraic structure (making it possible to detect singularities, such as self-intersections and cusps, that cannot be described solely in terms of the underlying topology). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「surface」の詳細全文を読む スポンサード リンク
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