The surjectivity and the continuity of definable functions in some definably complete locally o-minimal expansions and the Grothendieck ring of almost o-minimal structures

Abstract

In this paper, we first show that in a definably complete locally o-minimal expansion of an ordered abelian group (M, <, +, 0, ...) and for a definable subset X ⊆ M n which is closed and bounded in the last coordinate such that the set π n-1 (X) is open, the mapping π n-1 is surjective from X to M n-1 , where π n-1 denotes the coordinate projection onto the first n -1 coordinates.Afterwards, we state some of its consequences.Also we show that the Grothendieck ring of an almost o-minimal expansion of an ordered divisible abelian group which is not o-minimal is null.Finally, we study the continuity of the derivative of a given definable function in some ordered structures.