The complexity of being monitorable

We study monitorable sets from a topological standpoint. In particular, we use descriptive set theory to describe the complexity of the family of monitorable sets in a countable space $X$. When $X$ is second countable, we observe that the family of monitorable sets is ${\mathrm{\Pi }}_3^0$ and determine the exact complexities it can have. In contrast, we show that if $X$ is not second countable then the family of monitorable sets can be much more complex, giving an example where it is ${\mathrm{\Pi }}_1^1$-complete.