Year: 2018
Type: article
Abstract: We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are “tame” in several ways: (a) they are Hausdorff, (b) every point has a neighborhood which is definably homeomorphic to a definable set, (c) definable functions are piecewise continuous, (d) definable subsets have finitely many definably connected components, and (e) the frontier of a definable subset has lower dimension than the subset itself.
Source: Journal of Symbolic Logic
DOI: 10.1017/jsl.2018.50
Author: WILL JOHNSON
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2906562274