# Theories where any definable infinite set has interior

## Alf Dolich

### The City University of New York

We consider the case of a theory $T$ which expands that of divisible ordered Abelian groups and has the property that in any model of $T$ any infinite definable subset has non-empty interior (in the order topology). We will call such theories visceral. Visceral theories arise naturally in the study theories with a strong form of the independence property and generalize the class of o-minimal and weakly o-minimal theories. We consider the structure of definable sets and definable functions in visceral theories, giving some weak structural results. We also consider how to build general classes of examples of visceral theories, and relate these example back to questions about strong forms of the independence property.

Professor Dolich (Ph.D. 2002 University of Maryland, M.A. Columbia University, B.A. University of Pennsylvania) held a VIGRE Van Vleck Assistant Professorship at the University of Wisconsin, Madison, before coming to the New York area, where he now holds an Assistant Professor position at Kingsborough CC of CUNY. Professor Dolich conducts research in model theory, simple theories, and o-minimal theories with secondary interests in algebraic geometry and set theory.