The class of Borel sets is one of the most fundamental structures on a topological space. Its study lies at the intersection of several areas of mathematics; in this talk, we will investigate properties of the Borel algebra from the viewpoint of descriptive set theory and topology, focusing on the length of this hierarchy on a given second-countable space $X$. The length $ord(X)$ of the hierarchy is defined as the least ordinal $\alpha$ for which every Borel subset of $X$ is $\Sigma^0_\alpha$. The exact value of this ordinal turns out to be highly malleable, and a sophisticated forcing technique was developed by Arnold Miller to produce models of set theory in which it takes on arbitrary values. We will discuss the basic building blocks of this technique, as well as sketch the nature of rank arguments that yield consistency results about assignments of $ord(X)$ to several spaces $X$ simultaneously. Time permitting, we will also delve into the speaker’s recent contributions to this area, such as an extension of the framework to the study of generalized Borel hierarchies on topological spaces of uncountable weight.

https://arxiv.org/abs/2603.07377