On the first Banach problem concerning condensations of absolute κ-Borel sets onto compacta.

It is consistent that the continuum be arbitrary large and no absolute κ -Borel set X of density κ , ℵ 1 < κ < c ,condenses onto a compactum. It is consistent that the continuum be arbitrary large and any absolute κ -Borel set X of density κ , κ ≤ c , containing a closed subspace of the Baire space of weight κ , condenses onto a compactum. In particular, applying Brian's results in model theory, we get the following unexpected result. Given any A ⊆ N with 1 ∈ A , there is a forcing extension in which every absolute ℵ n -Borel set, containing a closed subspace of the Baire space of weight ℵ n , condenses onto a compactum if and only if n ∈ A . [ABSTRACT FROM AUTHOR]

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