Metric space/Ball/Definition

Open ball

Let ${}(M,d)$ be a metric space, ${}x\in M$, and ${}\epsilon >0$ a positive real number. Then

${}U{\left(x,\epsilon \right)}={\left\{y\in M\mid d(x,y)<\epsilon \right\}}\,$

is called the open ball, and

${}B\left(x,\epsilon \right)={\left\{y\in M\mid d(x,y)\leq \epsilon \right\}}\,$

is called the closed ball around ${}x$ of radius ${}\epsilon$.