Submission date: 22 November 2019

Abstract:

Comments: This work is a merger of arXiv:1404.3175 and the preprint Topologizing interpretable sets in o-minimal structures, among other things. Journal-ref: J. Symbolic Logic 83 (2018) 1477-1500 DOI: 10.1017/jsl.2018.50

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.

Mathematics Subject Classification: 03C64

Keywords and phrases:

Full text arXiv 1911.10077: pdf, ps.