| Open Ball |
Website Links For Ball |
Information AboutOpen Ball |
|
METRIC SPACES Let ''M'' be a Metric Space . The (open) ball of radius ''r'' > 0 centred at a point ''p'' in ''M'' is defined as : where ''d'' is the Distance Function or metric. If the less-than symbol (<) is replaced by a less-than-or-equal-to (≤), the above definition becomes that of a closed ball: :. Note in particular that a ball (open or closed) always includes p itself, since r > 0. 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 Set s 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 ''D''''n''; its outside is the ''n''-1-sphere ''S''''n''-1, e.g. the 3-sphere ''S''''3'' is the outside of ''D''''4'' in 4D. These and the corresponding objects in even higher dimensions are called hyperball and Hypersphere . See the latter for "volumes" and "areas". With other metrics the shape of a ball can be different; examples:
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 ; if it is smooth, it need not be Diffeomorphic to the Euclidean ball. SEE ALSO |
|
|