Definable Open Sets As Finite Unions of Definable Open Cells
Simon Andrews
Source: Notre Dame J. Formal Logic Volume 51, Number 2
(2010), 247-251.
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.
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03C64
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1276284785
Digital Object Identifier: doi:10.1215/00294527-2010-015
Zentralblatt MATH identifier: 05758440
Mathematical Reviews number (MathSciNet): MR2667935
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.
Mathematical Reviews (MathSciNet): MR1633348
Zentralblatt MATH: 0953.03045
[2] Edmundo, M., "Coverings by open cells in nonlinear o-minimal expansions of groups", Preprint, 2008.
Mathematical Reviews (MathSciNet): MR2395801
Zentralblatt MATH: 1135.03012
[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.
Mathematical Reviews (MathSciNet): MR1643405
Zentralblatt MATH: 0904.03021
Digital Object Identifier: doi:10.1112/S0024611598000549
[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.
Notre Dame Journal of Formal Logic
