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 -cells in weakly o-minimal structures, and prove that every weakly o-minimal structure with strong cell decomposition has -cell decomposition if and only if its canonical o-minimal extension has -cell decomposition. Then, we show that every weakly o-minimal structure with -cell decomposition satisfies . Our last result implies that every o-minimal structure in which every definable open set is a union of finitely many open -cells, has -cell decomposition.
"SCE-Cell Decomposition and OCP in Weakly O-Minimal Structures." Notre Dame J. Formal Logic 57 (3) 399 - 410, 2016. https://doi.org/10.1215/00294527-3507270