Uniform local weak o-minimality and $\ast$-local weak o-minimality

Abstract

We propose the notions of uniform local weak o-minimality and $\ast$-local weak o-minimality. Local monotonicity theorems hold in definably complete locally o-minimal structures and uniformly locally o-minimal structures of the second kind. In this paper, we demonstrate new local monotonicity theorems for uniformly locally weakly o-minimal structures of the second kind and for locally o-minimal structures under the assumption called the univariate $\ast$-continuity property. We also prove that several formulas for dimension of definable sets which hold in definably complete locally o-minimal structures also hold in $\ast$-locally weakly o-minimal structures possessing the univariate $\ast$-continuity property.