The Journal of Symbolic Logic

Research Article

An open mapping theorem for o-minimal structures

Joseph Johns

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

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 VRn be a definable open set and suppose that f: VRn is 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 = (a 1, b 1) × … × (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: Rn R n 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].

(Received October 15 1999)

(Revised September 24 2000)