Patrick Speissegger
Let $\M$ be an o-minimal expansion of a dense linear order $(M,\lt)$, and let $n \in \NN$. We now prove the cell decomposition theorem by induction on $n$, assuming that $n \ge 2$ and that (I)$_{n-1}$ and (II)$_{n-1}$ hold. Proof of (I)$_n$. As in the proof of the decomposition theorem for planar sets, we may…
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 $n \in \NN$ be nonzero. Inspired by this post, we now make the following Definition Let $\sigma \in \{0,1\}^n$ and set $\sigma’:= \sigma \rest{\{0,1\}^{n-1}}$. We say that a definable set $C \subseteq M^n$ is a $\sigma$-cell whenever the following holds: if…
Assume $\mathcal M$ and $S$ satisfy all of the assumptions in the exercise, including 1-4. For each $m\in\mathbb N$, define $A_m = \{x\in\Pi_n(S) : |S_x| \geq m\}$. We will show that each $A_m$ is either empty or equal to $S$, and hence that $S_x$ is equal to the minimal value of $m$ such that $A_m\neq\emptyset$.…
First some terminology: Let $X$ be a set and $Y_1, \dots, Y_l \subseteq X$, and put $\Y:= \{Y_1, \dots, Y_l\}$. We say that the $\Y$ partitions $X$ if $X = Y_1 \cup \cdots \cup Y_l$ and the $Y_j$-s are pairwise disjoint. Given $Z \subseteq X$, we say that $\Y$ is compatible with $Z$ if, for…