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!


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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.