Open ball

The set of points within distance < r of a center point in a metric space.

Open ball

Let $(X,d)$ be a metric space , let $x\in X$ , and let $r>0$ . The open ball of radius $r$ centered at $x$ is

B(x,r):=\{y\in X : d(x,y)<r\}.

Open balls are the basic building blocks of the topology induced by a metric : open sets are exactly those that contain an open ball around each of their points.

Examples:

In $\mathbb{R}$ with $d(x,y)=|x-y|$ , $B(a,r)=(a-r,a+r)$ .
In $\mathbb{R}^2$ with Euclidean distance, $B(0,1)$ is the open unit disk.
In a discrete metric space, $B(x,1)=\{x\}$ for every $x$ .