We introduce CE-cell decomposition, a modified version of the usual o-minimal cell decomposition. We show that if an o-minimal structure admits CE-cell decomposition then any definable open set in may be expressed as a finite union of definable open cells. The dense linear ordering and linear o-minimal expansions of ordered abelian groups are examples of such structures.
"Definable Open Sets As Finite Unions of Definable Open Cells." Notre Dame J. Formal Logic 51 (2) 247 - 251, 2010. https://doi.org/10.1215/00294527-2010-015