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 intersects each -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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-1-6,
author = {Ramiro de la Vega},
title = {Decompositions of the plane and the size of the continuum},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {65-74},
zbl = {1168.03038},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-1-6}
}
Ramiro de la Vega. Decompositions of the plane and the size of the continuum. Fundamenta Mathematicae, Tome 205 (2009) pp. 65-74. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-1-6/