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 2 2 κ has a dense subspace of size κ which is almost irresolvable and whose dispersion character is equal to κ.’’ In this paper we prove:(1) ) is equivalent to the existence of a dense subspace of 2 2 κ which is Baire submaximal and whose cardinality and dispersion character are both equal to κ. In particular, ( ⋆ κ ) implies that κ is measurable in an inner model of ZFC. Continuum Hypothesis holds, ( ⋆ κ ) fails for all κ. ) is equivalent to the existence of an ω 1 -complete ideal I on κ containing all sets of cardinality <κ and such that the quotient Boolean algebra P ( κ ) / I is isomorphic to the complete Boolean algebra that adjoins 2 κ Cohen reals.