## Dimension

Let $\M$ be an o-minimal expansion of a dense linear order $(M,\lt)$ with language $\la$. Definition Let $A \subseteq M$ and $\phi(x_1, \dots, x_n)$ be an $\la(A)$-formula, and set $S:= \phi(M^n)$. We define $$\dim_A S := \sup\set{\dim(s/A):\ \N \models \phi(s), \ \N \succ \M},$$ where $\dim(s/A)$ is defined in $\N$ as in this post. Note…

## Definable closure

Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. For $A \subseteq M$, the definable closure of $A$ is defined by $$\dcl(A):= \set{b \in M:\ \{b\} \text{ is } A\text{-definable}}.$$ Exercise Let $A \subseteq M$. Prove that $\dcl(A) = \acl(A)$. Let $\phi(x)$ be a formula with parameters in $A$. Prove that…