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 $\mathrm{CE}$-cells in weakly o-minimal structures, and prove that every weakly o-minimal structure with strong cell decomposition has $\mathrm{SCE}$-cell decomposition if and only if its canonical o-minimal extension has $\mathrm{CE}$-cell decomposition. Then, we show that every weakly o-minimal structure with $\mathrm{SCE}$-cell decomposition satisfies $\mathrm{OCP}$. Our last result implies that every o-minimal structure in which every definable open set is a union of finitely many open $\mathrm{CE}$-cells, has $\mathrm{CE}$-cell decomposition.