2015/08/30 by Pablo Cubides Kovacsics, Cubides-Kovacsics, Pablo, Luck Darnière +3
Mathematics · Computer Science · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1508.07536
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.