On Disjoint Borel Uniformizations

Abstract

Larman showed (1973, Mathematika20, 233–246) that any closed subset of the plane with uncountable vertical cross-sections has ℵ1 disjoint Borel uniformizing sets. Here we show that Larman's result is best possible: there exist closed sets with uncountable cross sections which do not have more than ℵ1 disjoint Borel uniformizations, even if the continuum is much larger than ℵ1. This negatively answers some questions of Mauldin (1990, “Open Problems in Topology” (J. van Mill and G. M. Reed, Eds.), 617–629). The proof is based on a result of Stern, stating that certain Borel sets cannot be written as a small union of low-level Borel sets (1978, C. R. Acad. Sci. Pan's Ser. A–B286, A855–A857). The proof of the latter result uses Steel's method of forcing with tagged trees (1978, Ann. Math. Logic15, 55–74); a full presentation of this method, written in terms of Baire category rather than forcing, is given here.