Ball mathematics

Contents

1.1 Examples
1.2 General definition
1.3 Related notions
1.4 See also

Geometry

In metric geometry, a ball is a set containing all points within a specified distance of a given point.

Examples

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.

General definition

Let M be a metric space. The (open) ball of radius r > 0 centred at a point p in M is defined as

$B_r(p) = \{ x \in M \mid d(x,p) < r \},$

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:

${\bar B}_r(p) = \{ x \in M \mid d(x,p) \le r \}$ .

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.

In n-dimensional Euclidean space, a closed unit ball is also denoted Dⁿ.

Related notions

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.

Topology

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 Bⁿ or Dⁿ. 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.