Past exam of the options II course of the University of Oxford / 2009 / E / E3 / b / v / 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, areIn 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 $d(a,\varnothing )=+\infty$.