## 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…