On splitting infinite-fold covers
Volume 212 / 2011
Fundamenta Mathematicae 212 (2011), 95-127
MSC: Primary 03E05, 03E15;
Secondary 03C25, 03E04, 03E35, 03E40, 03E50, 03E65, 05C15, 06A05,
52A20, 52B11.
DOI: 10.4064/fm212-2-1
Abstract
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.
