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 $Π^0_3$ 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 $ Π^1_1$-complete.

翻译：我们从拓扑学角度研究可监测集。特别地，我们运用描述集合论来描述可数空间$X$中可监测集族的复杂性。当$X$是第二可数空间时，我们观察到可监测集族属于$Π^0_3$类，并确定了其可能具有的确切复杂度。相反地，我们证明若$X$非第二可数，则可监测集族可能复杂得多，并通过实例说明其可达$Π^1_1$-完备的程度。