A theorem on restriction of a metric

A theorem on restriction of a metric

Consider $(X,d)$ be a metric space. Let $Y\subseteq X$ be a metric on
itself and $E\subseteq Y$. Then there is a theorem which says that $E$ is
open in $Y$ iff $E=Y\cap G$ for some open subset $G$ of the original space
$G$. The intuition behind behind this condition is not clear. Why is that
intersection is so special? I understand that $E$ being a subset of some
open set of $X$ is not enough because a subset of a open set need not be
an open set ?