# open ball of a limit point in a metric space

Suppose we are given a metric space $(X,d)$ with infinitely many elements whose set of isolated points $A$ in $X$ is finite.

Given a point $x \notin A$, we have that $x$ is a limit point. Does there exist some $r>0$ such that both $B_r(x)$ and $(B_r(x))^c$ are infinite?

Any help would be appreciated!

Cheers

• Can you find another point $y\ne x$ such that $y$ is a limit point? – John Douma Mar 28 '18 at 13:21
• @JohnDouma I can see how that's gonna help the argument but I'm not sure how to find such $y$. – user376127 Mar 28 '18 at 13:44
• How many points are not isolated? – John Douma Mar 28 '18 at 13:51
• @JohnDouma I see. Thanks! – user376127 Mar 29 '18 at 1:14