Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Fundamenta Mathematicae / Online First articles

Abstract

We show that separability and second-countability are first-order properties of topological spaces definable in o-minimal expansions of $(\mathbb R, \lt )$. We do so by introducing first-order characterizations – definable separability and definable second-countability – which make sense in a wider model-theoretic context. We prove that, within o-minimality, these notions have the desired properties, including their equivalence among definable metric spaces, and we conjecture a definable version of Urysohn’s Metrization Theorem.