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$S\subset\mathbb{R}^n$ is a set such that for each $x\in S$ there is a ball $B(x,r)$ such that $B(x,r)\cap S$ is countable.

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As a subset of $\Bbb R^n$, which is second countable, $S$ is second countable, too, thus Lindelöf. Now $S$ is covered by open balls $\{B_r(x)\cap S\mid x\in S\}$, so there is a subcovering of countably many such balls, each of which is countable.

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