Expansions of o-minimal structures by sparse sets

Harvey Friedman¹ ; Chris Miller¹

¹ Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, OH 43210, U.S.A.

Fundamenta Mathematicae, Tome 167 (2001) no. 1, pp. 55-64

Résumé

Given an o-minimal expansion $\mathfrak R$ of the ordered additive group of real numbers and $E\subseteq {\mathbb R}$, we consider the extent to which basic metric and topological properties of subsets of ${\mathbb R}$ definable in the expansion $({\mathfrak R},E)$ are inherited by the subsets of ${\mathbb R}$ definable in certain expansions of $({\mathfrak R},E)$. In particular, suppose that $f(E^m)$ has no interior for each $m\in {\mathbb N}$ and $f : {\mathbb R}^m\to {\mathbb R}$ definable in ${ \mathfrak R}$, and that every subset of ${\mathbb R}$ definable in $({\mathfrak R},E)$ has interior or is nowhere dense. Then every subset of ${ \mathbb R}$ definable in $({\mathfrak R}, (S))$ has interior or is nowhere dense, where $S$ ranges over all nonempty subsets of all cartesian products $E^k$ ($k \geq 1$). The same holds true with “nowhere dense” replaced by any of “null” (in the sense of Lebesgue), “countable”, “a finite union of discrete sets”, or “discrete”. We use this (together with a result of L. van den Dries) to give an example of an expansion of the real field that defines an isomorphic copy of the ordered ring of integers, yet does not define ${\mathbb Z}$.

Détail

DOI : 10.4064/fm167-1-4

Keywords: given o minimal expansion mathfrak ordered additive group real numbers subseteq mathbb consider extent which basic metric topological properties subsets mathbb definable expansion mathfrak inherited subsets mathbb definable certain expansions mathfrak particular suppose has interior each mathbb mathbb mathbb definable mathfrak every subset mathbb definable mathfrak has interior nowhere dense every subset mathbb definable mathfrak has interior nowhere dense where ranges nonempty subsets cartesian products geq holds nowhere dense replaced null sense lebesgue countable finite union discrete sets discrete together result nbsp van den dries example expansion real field defines isomorphic copy ordered ring integers yet does define mathbb