A Lower Density Operator for the Borel Algebra

Abstract

AbstractWe prove that the existence of a Borel lower density operator (a Borel lifting) with respect to the $$\sigma $$σ-ideal of countable sets, for an uncountable Polish space, is equivalent to CH. One of the implications is known (due to K. Musiał) and the remaining implication is derived from a general abstract result dealing with the negation of GCH. We observe that there is no lower density Borel operator with respect to the $$\sigma $$σ-ideal of countable sets, whose range is of bounded level in the Borel hierarchy.