Ai-maximal independent families and irresolvable Baire spaces

作者：Dorantes Aldama Alejandro; Pichardo Mendoza Roberto^*; Tamariz Mascarua Angel

来源：Topology and Its Applications, 2014, 175: 15-28.

DOI：10.1016/j.topol.2014.06.007

• Almost irresolvable spaces
• Baire spaces
• Boolean algebras
• Cantor cubes
• Ideals
• Independent families

A topological space is almost irresolvable if it cannot be written as a countable union of subsets with empty interior. Given a cardinal kappa, denote by (*kappa) the statement %26quot;the Cantor cube 2(2 kappa) has a dense subspace of size kappa which is almost irresolvable and whose dispersion character is equal to kappa.%26quot; In this paper we prove: %26lt;br%26gt;(1) (*kappa) is equivalent to the existence of a dense subspace of 2(2 kappa) which is Baire submaximal and whose cardinality and dispersion character are both equal to is. In particular, (*kappa) implies that kappa is measurable in an inner model of ZFC. %26lt;br%26gt;(2) If the Continuum Hypothesis holds, (*kappa) fails for all kappa. %26lt;br%26gt;(3) (*kappa) is equivalent to the existence of an omega(1)-complete ideal I on kappa containing all sets of cardinality %26lt; kappa and such that the quotient Boolean algebra P(kappa)/I is isomorphic to the complete Boolean algebra that adjoins 2(kappa) Cohen reals.

• 出版日期2014-9-15