Metric space/Ball/Definition

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

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

is called the open ball, and

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

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