= Solution
Every point other than $a$ is isolated, so no such point can enter the <closure> unless it was already in $B$. The only possible extra point is $a$, and it lies in the closure precisely when every $d$-ball about $a$ meets $B$. Therefore the three answers, in the printed order, are
$$
\boxed{\overline B=B;\qquad\overline B=B;\qquad\overline B=B\cup\{a\}}.
$$
In the middle case, a ball of radius below $d(a,B)$ misses $B$; in the last case the infimum zero makes every positive-radius ball meet $B$. If $B$ is empty its closure is empty as usual, with the convention $d(a,\varnothing)=+\infty$.
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