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

Notre Dame Journal of Formal Logic 57 (3):399-410 (2016)
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Abstract

Continuous extension cell decomposition in o-minimal structures was introduced by Simon Andrews to establish the open cell property 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.

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Weakly o-minimal nonvaluational structures.Roman Wencel - 2008 - Annals of Pure and Applied Logic 154 (3):139-162.
Definable Open Sets As Finite Unions of Definable Open Cells.Simon Andrews - 2010 - Notre Dame Journal of Formal Logic 51 (2):247-251.
Coverings by open cells.Mário J. Edmundo, Pantelis E. Eleftheriou & Luca Prelli - 2014 - Archive for Mathematical Logic 53 (3-4):307-325.

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