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An open mapping theorem for o-minimal structures

Published online by Cambridge University Press:  12 March 2014

Joseph Johns
Joseph Johns*
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago Illinois 60637, U.S.A., E-Mail: johns@math.uchicago.edu
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Extract

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 V ⊆ Rnbe a definable open set and suppose that f: V → Rnis 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 B ⊆ Rn is a Cartesian product of n definable open intervals: B = (a1, b1) × … × (an, bn) for some ai, bi, ∈ R ∪ {−∞, +∞}, with ai < bi, Given A ⊆ Rn, 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: Rn → Rn− 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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 4 , December 2001 , pp. 1817 - 1820
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

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