1 Ball

In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. more...

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Metric spaces

Let M be a metric space. The (open) ball of radius r > 0 centered at a point p in M is usually denoted by B_r(p) or B(p;r) and defined by

where d is the distance function or metric. 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 B_r or B and defined by:

Note in particular that a ball (open or closed) always includes p itself, since r > 0. Finally, the closure of the open ball B_r(p) is usually denoted

While it is always the case that not always the case that X with the discrete metric. In this case, for any B₁ = X, so clearly

A (open or closed) unit ball is a ball of radius 1.

A subset of a metric space is bounded 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 topology induced by d (by definition). This means, among other things, that all open sets in a metric space can be written as a union of open balls.

Euclidean balls

In n-dimensional Euclidean space with the ordinary (Euclidean) metric, if the space is the line, the ball is an interval, and if the space is the plane, the ball is the disc inside a circle. A closed unit ball is denoted by Dⁿ; its boundary (or \"edge\") is the n-1-sphere Sⁿ⁻¹, e.g., the 3-sphere S³ is the boundary of D⁴ in 4D. These and the corresponding objects in even higher dimensions are called hyperball and hypersphere. See the latter for \"volumes\" and \"areas\".

Read more at Wikipedia.org