Let $U\subset Y$ be open in $(Y,d_Y)$, where $d_Y:Y\times Y \rightarrow \mathbb{R}$ is the metric induced by $(X,d)$. Is $ U$ also open in $X$? Let $(X,d)$ be a metric space and $Y\subset X$.
Let $U\subset Y$ be open in $(Y,d_Y)$, where $d_Y:Y\times Y \rightarrow \mathbb{R}$ is the metric induced by $(X,d)$.
Is $ U$ then also inevitably open in $X$? If not, why?
Thoughts:
If $U$ is open in $(Y,d_Y)$ but not open in $(X,d)$ then there exists an $a\in U$ such that for every $r>0$:
$U_r(a)|_{d_Y}\subset U$ and $U_r(a)|_{d_X}\not\subset U$
with $U_r(a)|_{d_Y}=\{ y\in Y| d_Y(a,y)<r\}$ and $U_r(a)|_{d_X}=\{ x\in X| d(a,x)<r\}$

A necessary (and in fact sufficient) condition for every open subset of $Y$ to be open in $X$ is that $Y$ itself be open in $X.$
