Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 1 / a / ii / Solution

The closure $\overline{A}$ is the smallest closed subset of $M$ containing $A$, equivalently the intersection of all such closed subsets. In a metric space it is characterized byIf some ball misses $A$, its centre lies outside a closed superset of $A$; conversely the complement of any closed superset is an open neighbourhood disjoint from $A$. This verifies the equivalence of the two descriptions.