One Dimensional T.T.T Structures

TL;DR

This paper explores the topological and model-theoretic properties of one-dimensional t.t.t structures, establishing conditions under which they are o-minimal or locally o-minimal, and analyzing their connected components and group structures.

Contribution

It characterizes one-dimensional t.t.t structures in terms of o-minimality and topological assumptions, providing new insights into their structure and definability properties.

Findings

Structures split into multiple components when points are removed, indicating a simplex structure.

Additional assumptions ensure local o-minimality even without splitting.

Elementary extensions of saturated structures preserve the t.t.t property.

Abstract

In this paper we analyze the relationship between o-minimal structures and the notion of \omega -saturated one dimensional t.t.t structures. We prove that if removing any point from such a structure splits it into more than one definably connected component then it must be a one dimensional simplex of a finite number of o-minimal structures. In addition, we show that even if removing points doesn't split the structure, additional topological assumptions ensure that the structure is locally o-minimal. As a corollary we obtain the result that if an \omega -saturated one dimensional t.t.t structure admits a topological group structure then it is locally o-minimal. We also prove that the number of connected components in a definable family is uniformally bounded which implies that an elementary extension of an \omega -saturated one dimensional t.t.t structure is t.t.t as well.