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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
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 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 topologyTopology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaning study, talk. See also earth science, geography, human geography, g, 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 closureIn topology and mathematical analysis, the closure of a subset of a topological space is the smallest closed subset of which contains. This can be constructed by intersecting all closed supersets of in. Notation The closure of is written as or. If there i 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 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.
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