Notre Dame Journal of Formal Logic

Definable Open Sets As Finite Unions of Definable Open Cells

Simon Andrews

Abstract

We introduce CE-cell decomposition, a modified version of the usual o-minimal cell decomposition. We show that if an o-minimal structure $\mathcal{R}$ admits CE-cell decomposition then any definable open set in $\mathcal{R}$ may be expressed as a finite union of definable open cells. The dense linear ordering and linear o-minimal expansions of ordered abelian groups are examples of such structures.

Article information

Source
Notre Dame J. Formal Logic Volume 51, Number 2 (2010), 247-251.

Dates
First available in Project Euclid: 11 June 2010

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1276284785

Digital Object Identifier
doi:10.1215/00294527-2010-015

Mathematical Reviews number (MathSciNet)
MR2667935

Zentralblatt MATH identifier
05758440

Subjects
Primary: 03C64: Model theory of ordered structures; o-minimality

Keywords
o-minimal open cell property

Citation

Andrews, Simon. Definable Open Sets As Finite Unions of Definable Open Cells. Notre Dame J. Formal Logic 51 (2010), no. 2, 247--251. doi:10.1215/00294527-2010-015. http://projecteuclid.org/euclid.ndjfl/1276284785.


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References

  • [1] van den Dries, L., Tame Topology and O-Minimal Structures, vol. 248 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1998.
  • [2] Edmundo, M., "Coverings by open cells in nonlinear o-minimal expansions of groups", Preprint, 2008.
  • [3] Peterzil, Y., and S. Starchenko, "A trichotomy theorem for o-minimal structures", Proceedings of the London Mathematical Society. Third Series, vol. 77 (1998), pp. 481--523.
  • [4] Wilkie, A., "Covering definable open sets by open cells", in Proceedings of the RAAG Summer School Lisbon 2003: O-minimal Structures, edited by M. Edmundo, D. Richardson, and A. J. Wilkie, RAAG, Lisbon. 2005.