TITLE: Is there a definition of closed sets in terms of closed balls
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: 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 $C$.