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Disk (mathematics))

*A synonym for* ball *(in geometry* or *topology, and in any dimension) is* **disk** *(or* **disc***); however, a 3-dimensional ball is generally called a ball, and a 2-dimensional ball (e.g., the interior of a circle in the plane) is generally called a disk.*

## 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

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.

In *n*-dimensional Euclidean space, a closed unit ball is also denoted *D*^{n}.

### 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.

### See also

## Topology

In topology, *ball* has two meanings, with context governing which is meant.

The term **(open) ball** is informally used to refer to any open set: one speaks of "a ball about the point *p*" when one means an open set containing *p*. What this set is homemorphic to depends on the ambient space and on the open set chosen. Likewise, **closed ball** is used to mean the closure of such an open set. **Neighborhood** (or **neighbourhood**) is sometimes (and more properly) used instead of *ball*, although *neighborhood* also has a more general meaning: a neighborhood of *p* is any set *containing* an open set about *p*.

Also (and more formally), 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 *B*^{n} or *D*^{n}. 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.

### See also