Definable separability and second-countability in o-minimal structures

We show that separability and second-countability are first-order properties among topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$. 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 conjecture a definable version of Urysohn’s Metrization Theorem.