Restriction of a metric terminology

Question

If $(X,d)$ is a metric space and $Y\subseteq X$, does one say $Y$ is a metric space when the metric on $X$, $d$, is restricted to $Y$ or does one say $Y$ is a metric space when $d$ is restricted to $Y\times Y$ since the domain of $d$ is technically the product $X \times X$?