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# An open mapping theorem for o-minimal structures

Abstract

We fix an arbitrary o-minimal structure (R, ω, …), where (R, <) is a dense linearly ordered set without end points. In this paper “definable” means “definable with parameters from R”, We equip R with the interval topology and Rn with the induced product topology. The main result of this paper is the following.

Theorem. Let VRnbe a definable open set and suppose that f: VRnis a continuous injective definable map. Then f is open, that is, f(U) is open whenever U is an open subset of V.

Woerheide [6] proved the above theorem for o-minimal expansions of a real closed field using ideas of homology. The case of an arbitrary o-minimal structure remained an open problem, see [4] and [1]. In this paper we will give an elementary proof of the general case.

Basic definitions and notation. A box BRn is a Cartesian product of n definable open intervals: B = (a1, b1) × … × (an, bn) for some ai, bi, ∈ R ∪ {−∞, +∞}, with ai < bi, Given ARn, cl(A) denotes the closure of A, int(A) denotes the interior of A, bd(A) ≔ cl(A) − int(A) denotes the boundary of A, and ∂A ≔ cl(A) − A denotes the frontier of A, Finally, we let π: RnRn denote the projection map onto the first n − 1 coordinates.

Background material. Without mention we will use notions and facts discussed in [5] and [3]. We will also make use of the following result, which appears in [2].

References
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[1]Nesin A., Pillay A., and Razenj V., Groups of dimension two and three definable over o-minimal structures, Annals of Pure and Applied Logic, vol. 53 (1991), pp. 279296.
[2]Peterzil Y. and Steinhorn C., Definable compactness and definable subgroups of o-minimal groups, Journal of the London Mathematical Society, vol. 59 (1999), pp. 769786.
[3]Pillay A. and Steinhorn C., Definable sets in ordered structures. I, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565592.
[4]van den Dries L., o-Minimal structures, Logic: from foundations to applications (Hodges W. et al., editor), Oxford University Press, 1996.
[5]van den Dries L., Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, no. 248, Cambridge University Press, 1998.
[6]Woerheide A., Topology of definable sets in o-minimal structures, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1996.
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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic
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