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INTERPRETABLE SETS IN DENSE O-MINIMAL STRUCTURES

Published online by Cambridge University Press:  21 December 2018

WILL JOHNSON
WILL JOHNSON*
Affiliation:
NIANTIC, INC. 1 FERRY BUILDING SUITE 200 SAN FRANCISCO, CA, USAE-mail: willij6@gmail.com
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Abstract

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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.

Keywords

03C64o-minimalityinterpretable setsdefinable topologies
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 4 , December 2018 , pp. 1477 - 1500
Copyright
Copyright © The Association for Symbolic Logic 2018 

References

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