Tameness of definably complete locally o-minimal structures and definable bounded multiplication

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good-shaped definable subsets called quasi-special submanifolds. Using this fact, we investigate definably complete locally o-minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o-minimal expansions of ordered fields, Łojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo-definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo-definable spaces are definably compact. Here, a pseudo-definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set-valued functions with definably compact domains of definition.