Ball (mathematics) Guide, Meaning , Facts, Information and Description
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2 Topology |
In metric geometry, a ball is a set containing all points within a specified distance of a given point.
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 inside of a circle. With other metrics the shape of a ball can be different; for example, in taxicab geometry a ball is diamond-shaped.
Let M be a metric space. The (open) ball of radius r > 0 centred at a point p in M is defined as
In n-dimensional Euclidean space, a closed unit ball is also denoted Dn.
Open balls with respect to a metric d form a basis for the topology induced by d. This means, among other things, that all open sets in a metric space can be written as a union of open balls.
A set 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.
In topology, ball has two meanings, with context governing which is meant.
An (open) ball is any open set: one speaks of "a ball about the point p" when one means an open set containing p. What this ball is homemorphic to depends on the ambient space and on the ball chosen. A closed ball is the closure of a ball. Neighborhood (or neighbourhood) is sometimes used instead of ball, although neighborhood also has a more general meaning: a neighborhood of p is any set containing a ball about p.
Also, an (open or closed) ball is a space homeomorphic to the (open or closed) Euclidean ball described above under Geometry, but perhaps lacking its metric. A ball is known by its dimension: 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 smooth; if it is smooth, it need not be diffeomorphic to the Euclidean ball.Geometry
Examples
General definition
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.Related notions
See also
Topology