SCE-Cell Decomposition and OCP in Weakly O-Minimal Structures

Abstract

Continuous extension (CE) cell decomposition in o-minimal structures was introduced by Simon Andrews to establish the open cell property (OCP) in those structures. Here, we define strong CE-cells in weakly o-minimal structures, and prove that every weakly o-minimal structure with strong cell decomposition has SCE-cell decomposition if and only if its canonical o-minimal extension has CE-cell decomposition. Then, we show that every weakly o-minimal structure with SCE-cell decomposition satisfies OCP. Our last result implies that every o-minimal structure in which every definable open set is a union of finitely many open CE-cells, has CE-cell decomposition.