On the first Banach problem concerning condensations of absolute $$\kappa$$ -Borel sets onto compacta

Abstract

It is consistent that the continuum be arbitrary large and no absolute \(\kappa\)-Borel set X of density \(\kappa\), \(\aleph_1<\kappa<\mathfrak{c}\),condenses onto a compactum.

It is consistent that the continuum be arbitrary large and any absolute \(\kappa\)-Borel set X of density \(\kappa\), \(\kappa\leq\mathfrak{c}\), containing a closed subspace of the Baire space of weight \(\kappa\), condenses onto a compactum.

In particular, applying Brian's results in model theory, we get the following unexpected result. Given any \(A\subseteq \mathbb{N}\) with \(1\in A\), there is a forcing extension in which every absolute \(\aleph_n\)-Borel set, containing a closed subspace of the Baire space of weight \(\aleph_n\), condenses onto a compactum if and only if \(n\in A\).