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Preprint Number 914
914. Pablo Cubides-Kovacsics (LPP), Luck Darnière (LAREMA), Eva Leenknegt
Topological cell decomposition and dimension theory in P-minimal fields
Submission date: 30 August 2015.
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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