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The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n -ball of radius R is R n V n , {\displaystyle R^{n}V_{n},} where V n {\displaystyle V_{n}} is the volume of the unit n -ball , the n -ball of radius 1 .
An (open or closed) n-dimensional topological ball of X is any subset of X which is homeomorphic to an (open or closed) Euclidean n-ball. Topological n-balls are important in combinatorial topology, as the building blocks of cell complexes. Any open topological n-ball is homeomorphic to the Cartesian space R n and to the open unit n-cube ...
The term hypersphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension + , which means that they cannot be easily visualized. The n {\displaystyle n} -sphere is the setting for n {\displaystyle n} -dimensional spherical geometry .
A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n-ball (the solid ball in n dimensions), or the area of its surface, the n-sphere: being an n-dimensional figure, the volume scales as x n, while the surface area, being (n − 1)-dimensional, scales as x n−1.
A sphere or ball with unit radius and center at the origin of the space is called the unit sphere or the unit ball. Any arbitrary sphere can be transformed to the unit sphere by a combination of translation and scaling , so the study of spheres in general can often be reduced to the study of the unit sphere.
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A subset of R n is a null set if, for every ε > 0, it can be covered with countably many products of n intervals whose total volume is at most ε. All countable sets are null sets. If a subset of R n has Hausdorff dimension less than n then it is a null set with respect to n-dimensional Lebesgue measure.
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