TOPOLOGICALLY 1-BASED T-MINIMAL STRUCTURES

Source abstract

Abstract We prove group-existence and structure theorems in a general setting of tame topological theories. More precisely, we identify a linear/non-linear dividing line – called topological 1-basedness – among the class of t-minimal theories with the independent neighbourhood property. This is a wide class including all visceral theories, as well as all dense weakly o-minimal and C-minimal theories (even those where exchange fails). Now assume M $\mathcal M$ script upper M is highly saturated and t-minimal with the independent neighbourhood property. We show that if M $\mathcal M$ script upper M is non-trivial and topologically 1-based, it admits a type-definable abelian group ( G , + ) $(G,+)$ left parenthesis upper G comma plus right parenthesis with G an open subseteq of M . Moreover, we can ensure that G is a topological group with the subspace topology inherited from M ; and in this case, we show that the induced structure on G satisfies an appropriate topological analogue of the Hrushovski-Pillay classification of 1-based stable groups.