  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.
