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In mathematics, specifically in topology, a surface is a twodimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary threedimensional Euclidean space R^{3} — for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in threedimensional Euclidean space without introducing singularities or selfintersections.
To say that a surface is "twodimensional" means that, about each point, there is a coordinate patch on which a twodimensional coordinate system is defined. For example, the surface of the Earth is (ideally) a twodimensional sphere, and latitude and longitude provide coordinates on it.
Surfaces find application in physics, engineering, computer graphics, and many other disciplines, primarily when they represent 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.
Contents
Definitions and first examples
A (topological) surface is a nonempty second countable Hausdorff topological space on which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E^{2}. 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. This coordinates are known as local coordinates and these homeomorphisms lead us to describe surfaces as being locally Euclidean.
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