Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T06:05:16.121Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Denef, J. and van den Dries, L., 𝑝-adic and real subanalytic sets. The Annals of Mathematics, vol. 128 (1988), no. 1, pp. 79–138.CrossRefGoogle Scholar
Eleftheriou, P., Peterzil, Y., and Ramakrishnan, J., Interpretable groups are definable. Journal of Mathematical Logic, vol. 14 (2014), no. 1, p. 1450002.CrossRefGoogle Scholar
Fornasiero, A., Definable compactness for topological structures, unpublished, 2015.Google Scholar
Kamenkovich, S. and Peterzil, Y., Euler characteristic of imaginaries in o-minimal structures. Mathematical Logic Quarterly, vol. 63 (2017), no. 5, pp. 376–383.Google Scholar
Peterzil, Y., Constructing a group-interval in o-minimal structures. Journal of Pure and Applied Algebra, vol. 94 (1994), pp. 85–100.CrossRefGoogle Scholar
Peterzil, Y. and Steinhorn, C., Definable compactness and definable subgroups of o-minimal groups. Journal of the London Mathematical Society, vol. 59 (1999), pp. 769–786.CrossRefGoogle Scholar
van den Dries, L., Tame Topology and O-minimal Structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar