Give an example of a metric space $(X,d)$ where $B_{r}(x) = B_{r}[x]$ for every $x\in X$ and every $r\in\mathbb{R}_{>0}$.

Question

Exercise

Give an example of a metric space $(X,d)$ where $B_{r}(x) = B_{r}[x]$ for every $x\in X$ and every $r\in\mathbb{R}_{>0}$. Given another example where it does not occur.

My attempts

I have thought about the set of rationals $\mathbb{Q}$, but it fails to be the case, since we can have rational endpoints in an interval.

I have also tried to think about the discrete metric space, but it does not solve the problem neither.

Could anyone help me with this?

This is not homework. I am just studying the subject and I would like to understand it better.

Answer 1

If $x \neq y$ take $B_{d(x,y)}(x)$ If your space exists then $y \in B_{d(x,y)}(x)$ . Therefore $ d(x,y)< d(x,y)$ which is absurd. So the space must have only one point.