Open Ball

analysis topology

Definition

Open Ball

In a metric space $⟨M,d⟩$, the open ball with centre $x\in M$ and radius $r>0$ is the set

$$B_r(x)=\{y\in M\mid d(x,y)<r\}.$$

Intuition

An open ball contains exactly the points whose distance from the centre is strictly less than the radius.

Open balls generalise epsilon neighbourhoods to arbitrary metric spaces.