In a metric space $(X,d)$, by a closed ball I mean a set of the form $\{y: d(y,x)\le r\}$, where $r > 0$.
A common example where every open ball is a closed set and every closed ball is an open set is the ultrametric spaces: the metric spaces where $d(x,y)\le \max\{d(x,z),d(y,z)\}$. I would like to have an example where every open ball is a closed set but not every closed ball is an open set, and an example vice versa.
Definitions. An open ball is a set of the form $B_r(a) = \{x \mid d(x,a) < r\}$ where $r > 0$. A closed ball is a set of the form $\overline{B}_r(a) = \{x \mid d(x,a) \le r\}$ where $r > 0$.
Best Answer
Example 1. In the line, consider the set $X = \{0, 1, 1/2, 1/3, 1/4,\dots\}$ with the usual metric. I claim: every closed ball in $X$ is open in $X$; but the open ball ${B}_1(1)$ is not closed in $X$.