Ai-maximal independent families and irresolvable Baire spaces

A topological space is almost irresolvable if it cannot be written as a countable union of subsets with empty interior. Given a cardinal κ, denote by (⋆κ) the statement ''the Cantor cube 22κ has a dense subspace of size κ which is almost irresolvable and whose dispersion character is equal to κ.'' In this paper we prove: