March 1
Alf Dolich,
CUNY
Component Closed Structures on the Reals
A structure, R, expanding $(\mathbb{R}, \lt)$ is called component closed if whenever $X \subseteq R^n$ is definable so are all of $X$'s connected components. Two basic examples of component closed structures are $(\mathbb{R}, \lt, +, \cdot)$ and $(\mathbb{R}, \lt, \cdot, \mathbb{Z})$. It turns out that these two structures are exemplary of a general phenomenon for component closed structures from a broad class of expansions of $(\mathbb{R}, \lt)$: either their definable sets are very 'tame' (as in the case of the real closed field) or they are quite 'wild' (as in the case of the real field expanded by the integers).