Raadiuseball

Raadiuseball is a term used in mathematics to denote a metric ball—the set of points in a metric space whose distance from a fixed center does not exceed a specified radius. Formally, in a metric space (X, d), the closed raadiuseball of radius r centered at x is B̄(x, r) = { y in X | d(x, y) ≤ r }. The open raadiuseball is B(x, r) = { y in X | d(x, y) < r }.

In Euclidean space R^n with the usual distance, the closed raadiuseball is the closed disk (or ball) of radius r centered at x, while the open raadiuseball is the interior of that disk. The boundary of a closed raadiuseball is the sphere of radius r, denoted S(x, r) = { y | d(x, y) = r }. In most common spaces, raadiuseballs are bounded; in normed spaces they are convex when the norm is convex, which is always the case for standard norms.

The collection of raadiuseballs forms a basis for the metric topology: every open set is a union of open raadiuseballs. In analysis and topology, raadiuseball terminology is standard, with 'ball' sufficing in many texts; 'raadiuseball' is a nonstandard variant encountered in some educational resources or translations.

See also: Ball (geometry), Metric space, Open set, Closed set, Sphere.

References: standard texts on metric spaces and topology such as Munkres' Topology or Rudin's Real and Complex Analysis.