Decompositions of the plane and the size of the continuum

Ramiro de la Vega¹

¹ Universidad de los Andes Bogotá, Colombia

Fundamenta Mathematicae, Tome 203 (2009) no. 1, pp. 65-74

Résumé

We consider a triple $\langle E_0,E_1,E_2\rangle$ of equivalence relations on $\mathbb{R}^2$ and investigate the possibility of decomposing the plane into three sets $\mathbb{R}^2=S_0 \cup S_1 \cup S_2$ 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.

Détail

DOI : 10.4064/fm203-1-6

Keywords: consider triple langle rangle equivalence relations mathbb investigate possibility decomposing plane three sets mathbb cup cup each intersects each i class finitely many points many results literature starting famous theorem sierpi ski certain triples existence decomposition equivalent continuum hypothesis characterization zfc triples which decomposition exists application plane covered three sprays regardless size continuum answering question schmerl