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TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

Published online by Cambridge University Press:  21 March 2017

PABLO CUBIDES KOVACSICS
Affiliation:
LABORATOIRE NICOLAS ORESME UNIVERSITÉ DE CAEN CNRS U.M.R. 6139 F 14032 CAEN CEDEX, FRANCE E-mail: pablo.cubides@unicaen.fr
LUCK DARNIÈRE
Affiliation:
LAREMA, UNIVERSITÉ D’ANGERS 2 BD LAVOISIER, 49045 ANGERS CEDEX 01, FRANCE E-mail: luck.darniere@univ-angers.fr
EVA LEENKNEGT
Affiliation:
DEPARTMENT OF MATHEMATICS KULEUVEN CELESTIJNENLAAN 200B, 3001 HEVERLEE, BELGIUM E-mail: eva.leenknegt@wis.kuleuven.be

Abstract

This paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ 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 definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, 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.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2017 

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