The complexity of being monitorable

TL;DR

This paper analyzes the topological complexity of monitorable sets in countable spaces using descriptive set theory, revealing that their complexity varies significantly depending on whether the space is second countable.

Contribution

It characterizes the descriptive set-theoretic complexity of monitorable sets in countable spaces and distinguishes between second countable and non-second countable cases.

Findings

Monitorable sets are $oldsymbol{ ext{Pi}}^0_3$ in second countable spaces.

In non-second countable spaces, monitorable sets can be $oldsymbol{ ext{Pi}}^1_1$-complete.

The complexity of monitorable sets depends on the topological properties of the space.

Abstract

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 $\Pi^0_3$ 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 $ \Pi^1_1$-complete.