open ball
An open ball is the set of all points lying strictly within some fixed distance of a chosen center — the metric-space version of a disk drawn around a point, with the rim itself not included. It is the basic unit of nearness: to say a property holds near a point usually means it holds throughout some open ball around it.
Given a metric space (X, d), a center a in X and a radius r > 0, the open ball is B(a, r) = { x in X : d(x, a) < r }. The inequality is strict, so points sitting exactly at distance r are excluded. Open balls are the building blocks of the metric topology: a set is open precisely when it can be written as a union of open balls.
The shape of an open ball depends entirely on the metric, and can be wildly unintuitive. Under the taxicab metric on the plane the unit ball is a diamond; under the maximum metric it is a square; under the discrete metric the ball B(a, 1) contains only the single point a, while B(a, 2) is the whole space.
On the real line with d(x, y) = |x - y|, the open ball B(0, 1) is exactly the open interval (-1, 1). Open balls on R are nothing but open intervals centered at their center.
On the line, open balls are open intervals.