Definable Urysohn Metrization Conjecture

Let $(X,\tau)$ be a definable topological space in an o-minimal expansion of an ordered field. Call $(X,\tau)$ affine if it is definably homeomorphic to a set with the o-minimal Euclidean topology. Definable Urysohn Metrization Conjecture. The space $(X,\tau)$ is affine if and only if it is Hausdorff, regular, and definably second-countable.

This conjecture proposes that, in o-minimal expansions of ordered fields, Hausdorffness, regularity, and definable second-countability are exactly the conditions needed for a definable topological space to be Euclidean up to definable homeomorphism. The paper gives examples showing that definable second-countability is a genuine obstruction to affineness, but the proposed characterization remains open.