A Metric Space where the Closure of an Open Ball is not the Closed Ball

Last updated Nov 1, 2022

Consider $\mathbb{Z}$ equipped with the Discrete Metric. Then $\text{cl}B_{1}(0) \subsetneq \overline{B_{1}(0)}$.

By definition of Open Ball and Closed Ball $$B_{1}(0) = {x \in \mathbb{Z} : d(0,x) < 1} = {0}$$ and $$\overline{B_{1}(0)} = {x \in \mathbb{Z} : d(0,x) \leq 1} = \mathbb{Z}.$$ But since the Discrete Metric induces the Discrete Topology, we know $B_{1}(0)$ is also Closed. Therefore $\text{cl}B_{1}(0) = B_{1}(0) \subsetneq \overline{B_{1}(0)}$. $\blacksquare$