Let $(X,d_X)$ 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 $d_Y(y_1,y_2)=d_X(y_1,y_2)$ for all $y_1,y_2\in Y$. Let $i:Y\to X$ be the inclusion map defined by $i(y)=y$. Then $i$ is an isometric embedding.