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

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