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A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a ...
Volumes of balls in dimensions 0 through 25; unit ball in red. In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An n-ball is a ball in an n-dimensional Euclidean space.
A -ball, an ordinary ball, is the interior of a sphere ( -sphere). A 4 {\displaystyle 4} - ball is the interior of a 3 -sphere , etc. Topological description
An unit ball is the region inside of a unit sphere, the set of points of distance less than 1 from the center. A sphere or ball with unit radius and center at the origin of the space is called the unit sphere or the unit ball.
The space M is called precompact or totally bounded if for every r > 0 there is a finite cover of M by open balls of radius r. Every totally bounded space is bounded. To see this, start with a finite cover by r-balls for some arbitrary r. Since the subset of M consisting of the centers of these balls is finite, it has finite diameter, say D.
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula.
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For example, the boundary of a ball in E n looks locally like E n-1 and this leads to the notion of the inductive dimension. While these notions agree on E n, they turn out to be different when one looks at more general spaces. A tesseract is an example of a four-dimensional object.