Ball (mathematics) - Citizendia

In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

1 Metric spaces
2 Euclidean balls
3 Topological balls
4 See also

Metric spaces

Let M be a metric space. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined The (open) ball of radius r > 0 centered at a point p in M is usually denoted by $B r (p)$ or $B (p; r)$ and defined by

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

where d is the distance function or metric. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. This is also called an (open) metric ball. If the less-than symbol (<) is replaced by a less-than-or-equal-to (≤), the above definition becomes that of a closed (metric) ball, which is denoted by $B r [p]$ or $B [p; r]$ and defined by:

$B_r[p] \triangleq \{ x \in M \mid d(x,p) \le r \},$

Note in particular that a ball (open or closed) always includes p itself, since r > 0. Finally, the closure of the open ball $B r (p)$ is usually denoted $\overline{ B_r(p) }$ . In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set

While it is always the case that $B_r(p) \subseteq \overline{ B_r(p) }$ and $B_r(p) \subseteq B_r[p]$ , it is not always the case that $\overline{ B_r(p) } = B_r[p]$ . For example, consider a nonempty metric space $X$ with the discrete metric. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " In this case, for any $p \in X$ , $\overline{B_1(p)} = \{p\}$ and $B 1 [p] = X$ , so clearly $\overline{B_1(p)} \neq B_1[p]$ for all points $p \in X$ .

A (open or closed) unit ball is a ball of radius 1. In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed

A subset of a metric space is bounded if it is contained in a ball. In Mathematical analysis and related areas of Mathematics, a set is called bounded, if it is in a certain sense of finite size A set is totally bounded if given any radius, it is covered by finitely many balls of that radius. In Topology and related branches of Mathematics, a totally bounded space is a space that can be covered by finitely many Subsets of any

Open balls with respect to a metric d form a basis for the topology induced by d (by definition). In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. This means, among other things, that all open sets in a metric space can be written as a union of open balls. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets

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. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the A closed unit ball is denoted by Dⁿ; its boundary (or "edge") is the n-1-sphere Sⁿ⁻¹, e. For a different notion of boundary related to Manifolds see that article g. , the 3-sphere S³ is the boundary of D⁴ in 4D. In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. These and the corresponding objects in even higher dimensions are called hyperball and hypersphere. In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. See the latter for "volumes" and "areas".

With other metrics the shape of a ball can be different; examples:

in 2D:
- with the 1-norm (i. e. , in taxicab geometry) a ball is a square with the diagonals parallel to the coordinate axes
- with the Chebyshev distance a ball is a square with the sides parallel to the coordinate axes

in 3D:
- with the 1-norm a ball is a regular octahedron with the body diagonals parallel to the coordinate axes
- with the Chebyshev distance a ball is a cube with the edges parallel to the coordinate axes. Taxicab geometry, considered by Hermann Minkowski in the 19th century is a form of Geometry in which the usual metric of Euclidean geometry In Mathematics, Chebyshev distance (or Tchebychev distance) or L∞ metric is a metric defined on a Vector space where An octahedron (plural octahedra is a Polyhedron with eight faces A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex.

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 homeomorphic to an (open or closed) Euclidean ball described above. Topological equivalence redirects here see also Topological equivalence (dynamical systems. A ball is known by its dimension: an n-dimensional ball is called an n-ball and denoted $B n$ or $D n$ . In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it 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. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable

Contents

Metric spaces

Euclidean balls

Topological balls

See also