Let κ > ω be a regular cardinal and λ > κ a cardinal. The following partition property is shown to be consistent relative to a supercompact cardinal: For any with unbounded and 1 < γ < κ there is an unbounded Y ∪ X with for any n < ω.
A combinatorial statement concerning ideals of countable subsets of ω is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Suslin Hypothesis, that all (ω, ω*)-gaps are Hausdorff, and that every coherent sequence on ω either almost includes or is orthogonal to some uncountable subset of ω.
We investigate whether an arbitrary base for a dense-in-itself topological space can be partitioned into two bases. We prove that every base for a Lindelöf topology can be partitioned into two bases while there exists a consistent example of a first-countable, 0-dimensional, Hausdorff space of size and weight which admits a point countable base without a partition to two bases.
Three sets occurring in functional analysis are shown to be of class PCA (also called ) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12].
The additivity spectrum of an ideal is the set of all regular cardinals such that there is an increasing chain with . We investigate which set of regular cardinals can be the additivity spectrum of certain ideals. Assume that or , where denotes the -ideal generated by the compact subsets of the Baire space , and is the ideal of the null sets. We show that if is a non-empty progressive set of uncountable regular cardinals and , then in some c.c.c generic extension of the...
We study several perfect set properties of the Baire space which follow from the Ramsey property . In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.
We study splitting, infinitely often equal (ioe) and refining families from the descriptive point of view, i.e. we try to characterize closed, Borel or analytic such families by proving perfect set theorems. We succeed for hereditary splitting families and for analytic countably ioe families. We construct several examples of small closed ioe and refining families.
Under Martin’s axiom, collapsing of the continuum by Sacks forcing is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if proving that under this hypothesis there are no small uncountable maximal antichains in . We also construct a partition of into perfect sets which is a maximal antichain in and show that -sets are exactly (subsets of) selectors of maximal antichains of perfect sets.
We investigate properties of permitted trigonometric thin sets and construct uncountable permitted sets under some set-theoretical assumptions.