Expansions of o-minimal structures by sparse sets
Tom 167 / 2001
Streszczenie
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 ${\mathbb Z}$.
