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Abstract:

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.

Mathematics Subject Classification: 03C64 (Primary), 54A05, 54D65, 54D70 (Secondary)

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Full text arXiv 2405.07114: pdf, ps.