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!
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