On splitting infinite-fold covers

Márton Elekes; Tamás Mátrai; Lajos Soukup

Fundamenta Mathematicae (2011)

  • Volume: 212, Issue: 2, page 95-127
  • ISSN: 0016-2736

Abstract

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

How to cite

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Márton Elekes, Tamás Mátrai, and Lajos Soukup. "On splitting infinite-fold covers." Fundamenta Mathematicae 212.2 (2011): 95-127. <http://eudml.org/doc/286244>.

@article{MártonElekes2011,
abstract = { Let X be a set, κ be a cardinal number and let ℋ be a family of subsets of X which covers each x ∈ X at least κ-fold. What assumptions can ensure that ℋ can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover ℋ: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ℝⁿ by polyhedra and by arbitrary convex sets. We focus on problems with κ infinite. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory. },
author = {Márton Elekes, Tamás Mátrai, Lajos Soukup},
journal = {Fundamenta Mathematicae},
keywords = {splitting infinite cover; coloring; convex set; linearly ordered set; closed set; Cohen model; continuum hypothesis; Martin's axiom},
language = {eng},
number = {2},
pages = {95-127},
title = {On splitting infinite-fold covers},
url = {http://eudml.org/doc/286244},
volume = {212},
year = {2011},
}

TY - JOUR
AU - Márton Elekes
AU - Tamás Mátrai
AU - Lajos Soukup
TI - On splitting infinite-fold covers
JO - Fundamenta Mathematicae
PY - 2011
VL - 212
IS - 2
SP - 95
EP - 127
AB - Let X be a set, κ be a cardinal number and let ℋ be a family of subsets of X which covers each x ∈ X at least κ-fold. What assumptions can ensure that ℋ can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover ℋ: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ℝⁿ by polyhedra and by arbitrary convex sets. We focus on problems with κ infinite. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.
LA - eng
KW - splitting infinite cover; coloring; convex set; linearly ordered set; closed set; Cohen model; continuum hypothesis; Martin's axiom
UR - http://eudml.org/doc/286244
ER -

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