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 (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.