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Ball (mathematics)

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	Topics Ball (mathematics) Topic Home In mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , a ball is the inside of a sphereSphere A sphere is a perfectly symmetrical geometrical object.... ; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. Metric spaces Let M be a metric spaceMetric space In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.... . The (open) ball of radius r > 0 centered at a point p in M is usually denoted by or and defined by where d is the distance function or metricMetric (mathematics) In mathematics a metric or distance function is a function which defines a distance between elements of a set.... . This is also called an (open) metric ball. If the less-than symbol (<) is replaced by a less-than-or-equal-to (=), the above definition becomes that of a closed (metric) ball, which is denoted by or and defined by: Note in particular that a ball (open or closed) always includes `p` itself, since `r` > 0. Finally, the closureClosure (mathematics) Summary In mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of ... of the open ball is usually denoted . While it is always the case that and , it is not always the case that . For example, consider a nonempty metric space with the discrete metric. In this case, for any , and , so clearly for all points . A (open or closed) unit ballUnit ball In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of ... is a ball of radius 1. A subset of a metric space is boundedBounded set In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite ... if it is contained in a ball. A set is totally bounded if given any radius, it is covered by finitely many balls of that radius. Open balls with respect to a metric `d` form a basis for the topologyTopological space Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity.... induced by `d` (by definition). This means, among other things, that all open setOpen set In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "cha... s in a metric space can be written as a unionUnion (set theory) In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that ... of open balls. Euclidean balls In n-dimensional Euclidean spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... with the ordinary (Euclidean) metricEuclidean distance In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between the two points that one ... , if the space is the line, the ball is an intervalInterval (mathematics) In elementary algebra, an interval is a set that contains every real number between two indicated numbers and possibly the t... , and if the space is the plane, the ball is the disc inside a circleCircle In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed poi... . A closed unit ball is denoted by Dⁿ; its boundaryBoundary (topology) In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both f... (or "edge") is the n-1-sphere S^n-1, e.g., the 3-sphere3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere.... S³ is the boundary of D⁴ in 4D. These and the corresponding objects in even higher dimensions are called hyperball and hypersphereHypersphere Overview In mathematics, a hypersphere is a sphere which has dimension 3 or higher.... . See the latter for "volumes" and "areas". With other metrics the shape of a ball can be different; examples: in 2D: with the 1-norm (i.e., in taxicab geometryTaxicab geometry Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric of Eu... ) a ball is a square with the diagonals parallel to the coordinate axes with the Chebyshev distanceChebyshev distance In mathematics, the Chebyshev distance, also known as chessboard distance, between two points p and q in Eucli... a ball is a square with the sides parallel to the coordinate axes in 3D: with the 1-norm a ball is a regular octahedronOctahedron An octahedron is a polyhedron with eight faces.... with the body diagonals parallel to the coordinate axes with the Chebyshev distance a ball is a cubeFacts About Cube A cube is a three-dimensional Platonic solid composed of six square faces, facets or sides, with three meeting at each ver... with the edges parallel to the coordinate axes. Topological balls One may talk about balls in any topological space, not necessarily induced by a metric. An (open or closed) ball in such a space is a set which is homeomorphic to an (open or closed) Euclidean ball described above. A ball is known by its dimensionDimension In common usage, a dimension is a parameter or measurement required to define the characteristics of an object—i.e.... : an n-dimensional ball is called an n-ball and denoted or . For distinct n and m, an n-ball is not homeomorphic to an m-ball. A ball need not be smoothDifferentiable manifold Informally, a differentiable manifold is a kind of topological space that is similar enough to Euclidean space to allow one ... ; if it is smooth, it need not be diffeomorphic to the Euclidean ball. See also BallFacts About Ball Balls are usually hollow and spherical but can be other shapes, such as ovoid or solid .... - ordinary meaning Disk (mathematics)Disk (mathematics) In geometry, a disk is the region in a plane contained by a circle.... SphereSphere A sphere is a perfectly symmetrical geometrical object.... 3-sphere3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere.... HypersphereHypersphere In mathematics, a hypersphere is a sphere which has dimension 3 or higher.... Alexander horned sphereAlexander horned sphere The Alexander horned sphere is one of the most famous pathological examples in mathematics discovered in 1924 by J.... ManifoldManifold Summary A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in...