On splitting infinite-fold covers

Márton Elekes¹ ; Tamás Mátrai¹ ; Lajos Soukup¹

¹ Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences P.O. Box 127, H-1364 Budapest, Hungary

Fundamenta Mathematicae, Tome 212 (2011) no. 2, pp. 95-127.

Résumé

Let $X$ be a set, $\kappa$ be a cardinal number and let ${\cal H}$ be a family of subsets of $X$ which covers each $x\in X$ at least $\kappa$-fold. What assumptions can ensure that ${\cal H}$ can be decomposed into $\kappa$ many disjoint subcovers?We examine this problem under various assumptions on the set $X$ and on the cover ${\cal H}$: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ${\mathbb R}^{n}$ by polyhedra and by arbitrary convex sets. We focus on problems with $\kappa$ infinite. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.

Détail

DOI : 10.4064/fm212-2-1

Keywords: set kappa cardinal number cal family subsets which covers each least kappa fold what assumptions ensure cal decomposed kappa many disjoint subcovers examine problem under various assumptions set cover nbsp cal among other situations consider covers topological spaces closed sets interval covers linearly ordered sets covers mathbb polyhedra arbitrary convex sets focus problems kappa infinite besides numerous positive negative results many questions turn out independent usual axioms set theory