O-minimality and definable topologies

From the outset, topology has played an important role in the study of o-minimal structures. The central focus has been on developing the theory of o-minimality as a framework for 'tame topology', built upon the natural and well-behaved underlying euclidean (order) topology. However, also vital in the development and wider application of o-minimality has been the study of topological objects that are definable in o-minimal structures. These include definable groups, with their natural definable manifold topology; the definable complex analytic spaces of Bakker--Brunebarbe--Tsimerman, which play a key role in applications to Hodge theory; and definable linear orders, function spaces and metric spaces. Our approach, over the past decade, has been to seek a more general understanding of the nature of topological spaces definable in o-minimal structures. This has led to various classification and embedding results, in particular for one-dimensional spaces, as well as analogues of topological properties suitable for the o-minimal setting (such as compactness, separability, first-countability etc.) and several applications. Some of the inspiration for these results comes from open conjectures in set-theoretic topology, although our methods are very much grounded in elementary o-minimal theory. This project is joint work with Andújar Guerrero, in part also with Walsberg, and is related to work carried out independently by Peterzil and Rosel.