Question

Prove or disaprove ,The closure of an open vall in a metric space.is the corresponding closed ball

Answer 1

Let $(X,d)$ be a metric space where $d$ is the discrete metric. Let $x\in X$. Consider the open ball of radius $1$ around $x$. It's closure is the singleton set $\{x\}$. But the closed ball of radius $1$ is $X$ by the definition of the discrete metric. So the assertion is false.