Abstract: O-minimality is a branch of model theory with roots in real algebraic geometry which provides a family of settings for "tame topology": flexible enough to include most functions used in homotopy theory but without pathologies such as space-filling curves.

I will introduce a model category of spaces based on the definable sets of any o-minimal structure. These model categories resemble the Serre--Quillen model structure on topological spaces but inherit technical advantages from their construction. At the same time, they provide a context in which to better understand the weak polytopes of Knebusch (generalized to the o-minimal setting by Piekosz). This talk is based on joint work with Johan Commelin.