Definable separability and second-countability in o-minimal structures

2024/05/12 by Pablo Andújar Guerrero, Guerrero, Pablo Andújar
Computer Science · Mathematics · #03C64 (Primary) #54A05 #54D65 #54D70 (Secondary) #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #Logic, Reasoning, and Knowledge

paper · pdf · doi:10.48550/arxiv.2405.07114

openalex publication_date 2024/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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