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

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 subset of M-n which is closed and bounded in the last coordinate such that the set pi(n-1)(X) is open, the mapping pi(n-1) is surjective from X to Mn-1, where pi(n-1) denotes the coordinate projection onto the first n - 1 coordinates. Afterwards, we state some of its conse-quences. 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.