Open ball

The set of points within a given radius of a center point in a metric space, using strict inequality.

Open ball

An open ball in a metric space $(X,d)$ is a set of the form

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

where $x\in X$ and $r>0$ .

Open balls are open sets in the metric-induced topology and they form a basis for that topology; in particular, they are the basic neighborhoods in metric spaces.

Examples:

In $(\mathbb{R},|\cdot|)$ , $B(x,r)=(x-r,x+r)$ .
In the discrete metric on $X$ , $B(x,1)=\{x\}$ , while $B(x,r)=X$ for any $r>1$ .