Decompositions of the plane and the size of the continuum

Ramiro de la Vega

Fundamenta Mathematicae (2009)

  • Volume: 203, Issue: 1, page 65-74
  • ISSN: 0016-2736

Abstract

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We consider a triple ⟨E₀,E₁,E₂⟩ of equivalence relations on ℝ² and investigate the possibility of decomposing the plane into three sets ℝ² = S₀ ∪ S₁ ∪ S₂ in such a way that each S i intersects each E i -class in finitely many points. Many results in the literature, starting with a famous theorem of Sierpiński, show that for certain triples the existence of such a decomposition is equivalent to the continuum hypothesis. We give a characterization in ZFC of the triples for which the decomposition exists. As an application we show that the plane can be covered by three sprays regardless of the size of the continuum, thus answering a question of J. H. Schmerl.

How to cite

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Ramiro de la Vega. "Decompositions of the plane and the size of the continuum." Fundamenta Mathematicae 203.1 (2009): 65-74. <http://eudml.org/doc/282610>.

@article{RamirodelaVega2009,
abstract = {We consider a triple ⟨E₀,E₁,E₂⟩ of equivalence relations on ℝ² and investigate the possibility of decomposing the plane into three sets ℝ² = S₀ ∪ S₁ ∪ S₂ in such a way that each $S_i$ intersects each $E_i$-class in finitely many points. Many results in the literature, starting with a famous theorem of Sierpiński, show that for certain triples the existence of such a decomposition is equivalent to the continuum hypothesis. We give a characterization in ZFC of the triples for which the decomposition exists. As an application we show that the plane can be covered by three sprays regardless of the size of the continuum, thus answering a question of J. H. Schmerl.},
author = {Ramiro de la Vega},
journal = {Fundamenta Mathematicae},
keywords = {continuum hypothesis; Sierpiński’s theorem; sprays},
language = {eng},
number = {1},
pages = {65-74},
title = {Decompositions of the plane and the size of the continuum},
url = {http://eudml.org/doc/282610},
volume = {203},
year = {2009},
}

TY - JOUR
AU - Ramiro de la Vega
TI - Decompositions of the plane and the size of the continuum
JO - Fundamenta Mathematicae
PY - 2009
VL - 203
IS - 1
SP - 65
EP - 74
AB - We consider a triple ⟨E₀,E₁,E₂⟩ of equivalence relations on ℝ² and investigate the possibility of decomposing the plane into three sets ℝ² = S₀ ∪ S₁ ∪ S₂ in such a way that each $S_i$ intersects each $E_i$-class in finitely many points. Many results in the literature, starting with a famous theorem of Sierpiński, show that for certain triples the existence of such a decomposition is equivalent to the continuum hypothesis. We give a characterization in ZFC of the triples for which the decomposition exists. As an application we show that the plane can be covered by three sprays regardless of the size of the continuum, thus answering a question of J. H. Schmerl.
LA - eng
KW - continuum hypothesis; Sierpiński’s theorem; sprays
UR - http://eudml.org/doc/282610
ER -

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