Coverings by open cells

“…It is easy to construct an open definable subset of M 2 that cannot be expressed as a finite union of definable open cells. Edmundo, Elefteriou, and Prelli in [2] extended the result of Wilkie for o-minimal expansions of an ordered group. Andrews in [1] introduced continuous extension cell decomposition (i.e., denoted by CE-cell decomposition) in o-minimal structures.…”

Section: Introductionmentioning

confidence: 91%

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

Eivazloo¹,

Tari²

2016

Notre Dame J. Formal Logic

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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 ominimal 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.

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“…Edmundo, Eleftheriou, and Prelli [3] show that if R does not define a global field structure then U is a finite union of open cells. Suppose R defines a global field structure.…”

Section: The Structurementioning

confidence: 99%