Patrick Speissegger
Let $\M$ be an o-minimal expansion of a dense linear order $(M,\lt)$, and let $n \in \NN$. Our goal is to prove the Cell Decomposition Theorem (Knight, Pillay and Steinhorn) (I)$_n$ Let $S_1, \dots, S_k \subseteq M^n$ be definable. Then there exists a cell decomposition $\C$ of $M^n$ compatible with each $S_i$. (II)$_n$ Let $f:S…
Let $\M$ be an o-minimal expansion of a dense linear order $(M,\lt)$, and let $S \subseteq M^2$. Our goal is to show that if $S$ is definable and sparse, then $S$ satisfies uniform finitess. For $z \in S$, we say that $\Pi_1\rest{S}$ is a homeomorphism at $z$ if there exists an open box $B \subseteq…
Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. The first big question about o-minimality is the following: is o-minimality an elementary property, that is, given ${\cal N} \equiv {\cal M}$, is ${\cal N}$ necessarily o-minimal? Lemma The following are equivalent: every ${\cal N} \equiv {\cal M}$ is o-minimal; for every…