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

Abstract

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 tame dimension theory and decomposition theorem into good-shaped definable subsets called quasi-special submanifolds. Using this fact, in the latter part of this paper, 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 extension theorem and affiness of psudo-definable spaces hold true for such structures under the extra assumption that the domains of definition and the psudo-definable spaces are definably compact. Here, a pseudo-definable space is a topological spaces having finite definable atlases. We also demonstrate Michael's selection theorem for definable set-valued functions with definably compact domains of definition.

References in corpus (3)

• Locally o-minimal structures with tame topological properties
• Almost o-minimal structures and $\mathfrak{X}$-structures
• Functions definable in definably complete uniformly locally o-minimal structure of the second kind

Cited by in corpus (6)

• Notes on definably complete locally o-minimal expansions of ordered groups
• Definable compactness in definably complete locally o-minimal structures
• Affiness of topological space definable in a definably complete uniformly locally o-minimal structure of the second kind
• Dimension and topology in transserial tame pairs
• Definable quotients in locally o-minimal structures
• Locally o-minimal open core