  We consider locally o-minimal structures possessing tame topological
properties shared by models of DCTC and uniformly locally o-minimal expansions
of the second kind of densely linearly ordered abelian groups. We derive basic
properties of dimension of a set definable in the structures including the
addition property, which is the dimension equality for definable maps whose
fibers are equi-dimensional. A decomposition theorem into quasi-special
submanifolds is also demonstrated.
