If $B(x, r)$ is closed in $S\subseteq \mathbb{R}$, is it closed in $\mathbb{R}$?

Question

Let $(\mathbb{R}, d)$ a metric space. Let $S \subseteq \mathbb{R}$.

I know that an open ball in $S$ is not necessarily an open ball in $\mathbb{R}$, but is it a closed ball in $S$ closed in $\mathbb{R}$?

Answer 1

No. Consider $S = (-1,1)$. Then $\overline{B_S}(0,1) = S$ which is not closed in $\mathbb{R}$.

Answer 2

No, take $d$ as the Euclidean metric in $\mathbb{R}$ and take $S = (0, 1)$. Now let $B$ be large enough such that $B = S$, then $B$ is closed in $S$, but $B$ is not closed in $\mathbb{R}$.