The structural study of the Borel hierarchy on topological spaces is a foundational goal of descriptive set theory. By an early result of the field, we know that there exist universal sets at each level \(\alpha < \omega_1\) of the Borel hierarchy on the Baire space \(\omega^\omega\), hence the order of this hierarchy, i.e. the first ordinal \(\alpha\) at which every Borel set has been generated, attains the maximal possible value of \(\omega_1\). However, there are other subspaces of \(\omega^\omega\) where this hierarchy is shorter; take for example any countable space, on which every Borel subset must be \(\Sigma^0_2\).

The topic of this talk is a framework for the surgical alteration of the complexity of the Borel hierarchy on subspaces of \(\omega^\omega\), pioneered by A. Miller. We will discuss Miller's notion of \(\alpha\)-forcing, which allows for either collapsing or increasing the length of the Borel hierarchy, as well as the proof ideas behind some preservation theorems necessary to do so. In the second part of the talk, we will delve into recent developments in this area, such as an extension of the framework to the field of generalized descriptive set theory of an uncountable cardinal \(\kappa = \kappa^{<\kappa}\) or the study of the \(\lambda\)-Borel subsets of \(\kappa^\kappa\) for \(\lambda > \kappa\), with a particular emphasis on the case \(\lambda = \omega_1\) and \(\kappa = \omega\). We will give several examples of models constructed using this method in both the classical case of \(\omega\) and the generalized case of an uncountable \(\kappa\). Lastly, we will discuss some limitations of the technique and directions for future work.