$B\in\Bbb R$ has the property that given $b∈B$ there exists $k>0$ such that if $0<|b−x|A subset $B$ of $\mathbb R$ has the property that given $b ∈ B$ there exists $k > 0$ such that if $0 < |b − x| < k$ for some $x ∈ \mathbb R$, then $x \notin B$. Is $B$ countable?
I tried using the diagonal argument here to show that $B$ is uncountable but I can't seem to make much progress...

The hypothesis clearly implies that $B$ has no limit points. By Bolzano - Weirstarss Theorem $B$ can have only finitely many points in $(-n,n)$ for any $n$. so $B$ is at most countable.
