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