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arXiv · 2405.07114

Definable separability and second-countability in o-minimal structures

Pablo Andújar Guerrero
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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 $(\mathbb{R},<)$. 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.

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Pablo Andújar Guerrero. 2024-05-12. Definable separability and second-countability in o-minimal structures. https://arxiv.org/abs/2405.07114

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