Definable separability and second-countability in o-minimal structures

TL;DR

This paper introduces first-order notions of separability and second-countability in o-minimal structures, proving their equivalence in definable metric spaces and proposing a conjecture related to Urysohn's Metrization Theorem.

Contribution

It develops first-order characterizations of key topological properties in o-minimal structures and establishes their fundamental properties within this framework.

Findings

Separable and second-countable properties are first-order in o-minimal structures.

These properties are equivalent in definable metric spaces.

A conjecture for a definable Urysohn's Metrization Theorem is proposed.

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.