  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.
