Balls in General Metric Spaces
Let (M,d) be a metric space, namely a set M with a metric (distance function) d. The open (metric) ball of radius r > 0 centered at a point p in M, usually denoted by Br(p) or B(p; r), is defined by
The closed (metric) ball, which may be denoted by Br or B, is defined by
Note in particular that a ball (open or closed) always includes p itself, since the definition requires r > 0.
The closure of the open ball Br(p) is usually denoted . While it is always the case that and, it is not always the case that . For example, in a metric space with the discrete metric, one has and, for any .
An (open or closed) unit ball is a ball of radius 1.
A subset of a metric space is bounded if it is contained in some ball. A set is totally bounded if, given any positive radius, it is covered by finitely many balls of that radius.
The open balls of a metric space are a basis for a topological space, whose open sets are all possible unions of open balls. This space is called the topology induced by the metric d.
Read more about this topic: Ball (mathematics)
Famous quotes containing the words balls, general and/or spaces:
“If the head is lost, all that perishes is the individual; if the balls are lost, all of human nature perishes.”
—François Rabelais (1494–1553)
“There was not a tree as far as we could see, and that was many miles each way, the general level of the upland being about the same everywhere. Even from the Atlantic side we overlooked the Bay, and saw to Manomet Point in Plymouth, and better from that side because it was the highest.”
—Henry David Thoreau (1817–1862)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)