Suppose $X$ is a metric space such that for each $x\in X$, $\exists \epsilon_x>0$ such that $\overline{B(x,\epsilon_x)}$ is compact.Then show that this metric space is complete.
This is a problem from Elements of Metric Spaces by M.N. Mukherjee.But I think this problem is wrong.The statement is not true.Suppose $X=(0,1)$ with $(x,y)\mapsto |x-y|$ as metric.Then every $X$ satisfies the given property but is not a complete metric space.But,can this problem be modified to get some interesting results.
I am looking for some interesting properties of the metric spaces $X$ which satisfy the property, for each $x\in X$, $\exists \epsilon_x>0$ such that $\overline{B(x,\epsilon_x)}$ is compact. Can someone give me some lead on this?
Your counterexample is just fine.
The spaces for which that property holds are locally compact. Therefore, by the Baire category theorem, it is a Baire space, that is, a space such that the intersection of countably many dense open sets is still dense.