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Friday, August 10, 2018

**Abstract:** In model theory a topological space can be interpreted naturally as a second order structure, namely a set and the unary relation of subsets that corresponds to the topology. A different approach is that of a first order structure in which there exists a topological space with a basis that is (uniformly) definable. We call this a definable topological space. A natural example corresponds to the order topology on any linearly ordered structure. During this talk I’ll present results on definable topological spaces in an o-minimal structure $\mathcal{R} = (R, ...)$. In particular we will classify Hausdorff definable topological spaces $(X, \tau )$ where $X \subseteq R$. We’ll consider a number of first order properties of these spaces that resemble topological properties and note how, in the o-minimal setting, the induced framework, which we might call “definable topology”, resembles general topology. This is joint work with Margaret Thomas and Erik Walsberg.