Definable one dimensional structures in o-minimal theories

摘要

This is the first of two papers where we prove the Zil’ber trichotomy principle for one-dimensional structures definable in o-minimal ones. Here we prove: Let N be a definable structure in an o-minimal structure M, with dim _M (N) = 1. If N is stable then it is necessarily 1-based. Along the way, we develop a fine intersection theory for definable curves in o-minimal structures