closed ball
A closed ball is the open ball together with its outer rim: all points at distance at most a fixed radius from the center, the boundary included. It is the natural model of a solid disk that contains its own edge.
Given a metric space (X, d), a center a and a radius r >= 0, the closed ball is B-bar(a, r) = { x in X : d(x, a) <= r }, with the inequality now non-strict. A closed ball is always a closed set, since its complement is open. It always contains the corresponding open ball, and the two differ only by the points lying exactly at distance r.
Be careful: the closed ball of radius r is not always the closure of the open ball of radius r. In the usual Euclidean spaces they do coincide, but in the discrete metric the open ball B(a, 1) = {a} is already closed, so its closure is {a}, whereas the closed ball of radius 1 is the entire space. The two notions agree only when the metric is well-behaved.
On R, the closed ball B-bar(0, 1) is the closed interval [-1, 1]; its endpoints 1 and -1 lie at distance exactly 1 from the center and so are included.
On the line, closed balls are closed intervals.