Topologically 1-based T-minimal Structures

TL;DR

This paper establishes a classification of t-minimal theories based on topological 1-basedness, proving the existence of type-definable abelian groups within these theories and extending stability results to a broader topological context.

Contribution

It introduces the concept of topological 1-basedness in t-minimal theories and proves the existence of type-definable abelian groups in such theories, extending stability classifications.

Findings

Topological 1-basedness divides t-minimal theories into distinct classes.

Non-trivial topologically 1-based t-minimal theories contain type-definable abelian groups.

The structure on these groups satisfies a topological analog of the Hrushovski-Pillay classification.

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 neighborhood 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 $\mathcal M$ is highly saturated and t-minimal with the independent neighborhood property. We show that if $\mathcal M$ is non-trivial and topologically 1-based, it admits a type-definable abelian group $(G,+)$ with $G$ an open subset 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 analog of the Hrushovski-Pillay classification of 1-based stable groups.