Manifolds With Boundary
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In mathematics, a manifold of dimension n is a topological space that near each point resembles n-dimensional Euclidean space. More precisely, each point of an n-dimensional manifold has a neighbourhood (mathematics)|neighbourhood that is homeomorphic to the Euclidean space of dimension n. Line (geometry)|Lines and circles, but not Lemniscate|figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the Plane (geometry)|plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot. Although near each point, a manifold resembles Euclidean space, globally a manifold might not. For example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of geographic maps: map projections of the region into the Euclidean plane. When a region appears in two neighbouring maps (in the context of manifolds they are called...
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