Metric space/Open subset/Epsilon/Definition

Let ${\displaystyle {}(M,d)}$ be a metric space. A subset ${\displaystyle {}U\subseteq M}$ is called open (in ${\displaystyle (M,d)}$), if for every ${\displaystyle {}x\in U}$ there exists an ${\displaystyle {}\epsilon >0}$ such that

${\displaystyle {}U{\left(x,\epsilon \right)}\subseteq U\,}$

holds.