D-MINIMAL EXPANSIONS OF THE REAL FIELD HAVE THE EXCHANGE PROPERTY*

CHRIS MILLER

Version: December 15, 2006

Throughout, ℜ denotes a structure on ℝ. Definability is with respect to ℜ. Given A ⊆ ℝ, dcl(A) denotes the definable closure of A, that is, the set of all A-definable points in ℝ.

ℜ has the Exchange Property if for all A ⊆ ℝ and x, y ∈ ℝ:

y ∈ dcl(A ∪ {x}) \ dcl(A) ⟹ x ∈ dcl(A ∪ {y}).

ℜ is d-minimal if for every n ∈ ℕ and definable A ⊆ ℝ^{n+1}, there exists N ∈ ℕ such that for every x ∈ ℝ^{m}, the fiber { t ∈ ℝ : (x, t) ∈ A } either has interior or is a union of N (not necessarily distinct) discrete sets. (Recall that Y ⊆ ℝ^{m} is discrete if for every y ∈ Y there is an open box B such that Y ∩ B = {y}.)

Proposition. If ℜ is a d-minimal expansion of (ℝ, +, ·), then it has the Exchange Property.

Proof. Let A ⊆ ℝ and x, y ∈ ℝ be such that y ∈ dcl(A ∪ {x}) \ dcl(A). Then there is an A-definable function f: ℝ → ℝ such that f(x) = y. By arguing as in the proof of [Mil05, Theorem 3.3], there is an A-definable S ⊆ ℝ such that S is closed and has no interior, and for every open interval I ⊆ ℝ \ S, the restriction f↾I is either constant or strictly monotone. By d-minimality, there exists N ∈ ℕ such that S is a union of N discrete sets. The set of all isolated points of S is A-definable. Since ℚ ⊆ dcl(∅), every isolated point of S is A-definable. Hence (by an easy induction on N), each s ∈ S is A-definable. Since y ∉ dcl(A), we have x ∉ S, so there is an A-definable open interval I about x such that f↾I is strictly monotone. Since the compositional inverse of f↾I is A-definable, x ∈ dcl(A ∪ {y}). □

An examination of the proof shows that we needed neither d-minimality nor the field structure per se:

Corollary (of the proof). Suppose that < is ∅-definable and dcl(∅) is dense in ℝ—in particular, if ℜ is an expansion of (ℝ, <, +, 1)—and every definable subset of ℝ either has interior or is a finite union of discrete sets. Then ℜ has the Exchange Property.

Warning. It may appear that no use was made of working over ℝ, but this is not true: I do not know how to get the conclusion of the third sentence of the proof just from d-minimality (I use the Baire Category Theorem).

References

[Mil05] C. Miller, Tameness in expansions of the real field, Logic Colloquium ’01 (Vienna, 2001), Lect. Notes in Logic, vol. 20, Assoc. Symbolic Logic, Urbana, IL, 2005, pp. 281–316.

Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210, USA

   E-mail address: miller@math.ohio-state.edu
   URL: http://www.math.ohio-state.edu/~miller

*This is not a preprint; please do not refer to it as such.