One Dimensional T.T.T Structures

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In this paper we analyze the relationship between o-minimal structures and the notion of ω-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 ω-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 ω-saturated one dimensional t.t.t structure is t.t.t as well.