Is there a definition of closed sets in terms of closed balls In my general topology lectures, closed sets are defined in terms of open sets but I wonder whether closed sets can be defined in terms of closed balls in metrisable topological spaces. 
 A: What about this:

A set $C$ is closed if around every point not in $C$, there exists a closed ball of non-zero radius which doesn't intersect with $C$.

Explanation:
A set is closed iff its complement is open. In the standard definition, its complement is open if it contains an open ball around every of its points. A closed ball has an open ball as subset iff its radius is not zero (namely the open ball with the same radius). Moreover, every open ball contains a closed ball of non-zero radius (for example, the closed ball with half the radius). Therefore there's an open ball in the complement of $C$ around a point iff there is a closed ball with non-zero radius in the complement of $C$ around that point. Of course a closed ball in the complement of $C$ is a closed ball that doesn't intersect with $C$.
A: It is true that in a metric space every closed set can be represented as the union of (possibly degenerate, or singleton) closed balls. However the arbitrary union of closed sets is not closed in the usual metric topology, or in general spaces. If we were to define a topology where the closed sets were the arbitrary union closed balls we would have so many closed sets that the topology would be discrete.
