Formal Topology in Univalent Foundations

Abstract

Formal topology is a mathematical discipline that aims to interpret topology in type-theoretical terms, that is, constructively and predicatively. Type theory has recently undergone a transformation [42] through insights arising from its association with homotopy theory, resulting in the formulation of the notion of a univalent type theory, and more broadly, a univalent foundation. We investigate, in this thesis, the natural continuation of the line of work on formal topology into univalent type theory. We first recapitulate our finding that a naive approach to formal topology in univalent type theory is problematic, and then present a solution to this problem that involves the use of higher inductive types. We hence sketch the beginnings of an approach towards developing formal topology in univalent type theory. As a proof of concept for this approach, we develop the formal topology of the Cantor space and construct a proof that it is compact. The development that we present has been carried out using the cubical extension of the Agda proof assistant [44]. No postulates have been used and the development typechecks with the --safe flag of Agda. The presentation in this thesis amounts to an informalisation of this formal development.