Let $(X,d)$ be a \reftext{def:metric-space-2026a}{metric space}, and let $U\subseteq X$. We say that $U$ is open in the metric space $(X,d)$ if for every point $x\in U$ there exists a real number $r>0$ such that

where $B_d(x,r)$ is the \reftext{def:open-ball-metric-space-2026a}{open ball} with center $x$ and radius $r$.