Formal Topology and Univalent Foundations

Book chapter, 2021

We present an approach to the representation of formal topology in univalent foundations. This question turns out to be quite rich and connects with basic questions of constructive mathematics, such as bar induction, impredicativity and inductive definitions. Specifically, we use a higher inductive type with a built-in truncation constructor to represent the inductively defined covering relation, which seems to be necessary to avoid choice principles when defining the cover in a predicative manner. The development that we present has been formally checked in the Cubical Agda proof assistant.