One Dimensional T.T.T Structures

The notion of a first order topological structure was introduced by Pillay [1] as a generalization of the notion of an o-minimal structure. The idea is to provide a general framework in which model theory can be used to analyze a topological structure whose topology isn’t necessarily induced by a definable order. In the o-minimal case, the topology is generated from a basis where each basis set can be defined by substituting the variables y1 and y2 by suitable parameters in the following formula φ(x, y1, y2) = y1 < x < y2 A first order topological structure generalizes this to the case where φ is some arbitrary formula with more than one variable. Pillay also introduced the notion of topologically totally transcendental (t.t.t) structures which have the additional property that any definable set has a finite number of connected components. For example, by definition o-minimal structures are t.t.t. In the previously mentioned paper, Pillay proved that one dimensional t.t.t structures have some characteristics in common with o-minimal structures such as the exchange property. Furthermore, he showed that if the topology of a one dimensional t.t.t structure is induced by a definable dense linear ordering then the structure is o-minimal. In this paper we’ll focus on ω-saturated one dimensional t.t.t structures and prove that under a few additional topological assumptions, such structures are composed of o-minimal components in a relatively simple manner. Our main result which will be proved in section 4 will be to show that if we assume that removing any point from the structure splits it into at least two connected components, then the structure must be a one dimensional simplex of a finite number of o-minimal structures: