Ai-maximal independent families and irresolvable Baire spaces

Abstract

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:(1)(⋆κ) is equivalent to the existence of a dense subspace of 22κ 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.(2)If the Continuum Hypothesis holds, (⋆κ) fails for all κ.(3)(⋆κ) is equivalent to the existence of an ω1-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 2κ Cohen reals.