April 24
Margaret Thomas,
Purdue University and Institute for Advanced Study
O-minimality and definable topologies
O-minimal structures, defined in the 1980s, have been extensively studied from a topological perspective. They are considered an excellent framework for `tame topology', given the well-behaved nature of their underlying euclidean (order) topology, and there have also been numerous studies of the topological nature of various definable objects in such structures (e.g. groups, manifold spaces, orders, function spaces and metric spaces). Our work has continued this second theme, towards a more general understanding of the nature of topological spaces definable in o-minimal structures. So far, our focus has mainly been on one-dimensional spaces but, even in this setting, there are examples exhibiting a wide variety of properties, including classical topological counterexamples. Nevertheless, we obtain a number of classification results for such spaces satisfying various classical separation axioms, in terms of decomposition and embedding theorems. This also requires us to find analogues of topological properties suitable for the o-minimal setting (such as compactness, separability, first-countability etc.) and leads to several applications, including definable versions of conjectures from classical set-theoretic topology due to Gruenhage and Fremlin. This is joint work with Andújar Guerrero (building on earlier joint work with Walsberg, and related to work carried out independently by Peterzil and Rosel).