Let $(X,d)$ be a [metric space](/page/Metric%20Space), let $Y\subset X$, and equip $Y$ with the subspace metric $d_Y:Y\times Y\to [0,\infty)$ defined by $d_Y(y_1,y_2)=d(y_1,y_2)$. For a subset $U\subset Y$, the set $U$ is open in the metric space $(Y,d_Y)$ if and only if there exists an [open set](/page/Open%20Set) $G\subset X$ in the metric space $(X,d)$ such that

\begin{align*} U=Y\cap G. \end{align*}