# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## On splitting infinite-fold covers

### Tom 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

#### Streszczenie

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.

#### Autorzy

• Márton ElekesAlfréd Rényi Institute of Mathematics
P.O. Box 127, H-1364 Budapest, Hungary
e-mail
e-mail
• Tamás MátraiAlfréd Rényi Institute of Mathematics
P.O. Box 127, H-1364 Budapest, Hungary
e-mail
e-mail
• Lajos SoukupAlfréd Rényi Institute of Mathematics