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Topological cell decomposition and dimension theory in P-minimal fields

Abstract : This paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of the frontier of A is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any m-ary definable function is continuous on a dense, relatively open subset of its domain, thereby answering a question that was originally posed by Haskell and Macpherson. In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.
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Submitted on : Saturday, August 29, 2015 - 12:04:44 AM
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Pablo Cubides-Kovacsics, Luck Darnière, Eva Leenknegt. Topological cell decomposition and dimension theory in P-minimal fields. The Journal of Symbolic Logic, Association for Symbolic Logic, 2017, 82 (1), pp.347-358. ⟨10.1017/jsl.2016.45⟩. ⟨hal-01188341⟩



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