Selection principles in mathematics: A milestone of open problems

We survey some of the major open problems involving selection principles, diagonalizations, and covering properties in topology and infinite combinatorics. Background details, definitions and motivations are also provided.Comment: Small update

The combinatorics of splittability

Marion Scheepers, in his studies of the combinatorics of open covers, introduced the property Split(U,V) asserting that a cover of type U can be split into two covers of type V. In the first part of this paper we give an almost complete classification of all properties of this form where U and V are significant families of covers which appear in the literature (namely, large covers, omega-covers, tau-covers, and gamma-covers), using combinatorial characterizations of these properties in terms related to ultrafilters on N. In the second part of the paper we consider the questions whether, given U and V, the property Split(U,V) is preserved under taking finite unions, arbitrary subsets, powers or products. Several interesting problems remain open.Comment: Small update

Some new directions in infinite-combinatorial topology

We give a light introduction to selection principles in topology, a young subfield of infinite-combinatorial topology. Emphasis is put on the modern approach to the problems it deals with. Recent results are described, and open problems are stated. Some results which do not appear elsewhere are also included, with proofs.Comment: Small update

On weakly tight families

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $\c < {\aleph}_{\omega}$, we construct a weakly tight family under the hypothesis \s \leq \b < {\aleph}_{\omega}. The case when \s < \b is handled in \ZFC and does not require \b < {\aleph}_{\omega}, while an additional PCF type hypothesis, which holds when \b < {\aleph}_{\omega} is used to treat the case \s = \b. The notion of a weakly tight family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hru{\v{s}}{\'a}k and Garc{\'{\i}}a Ferreira \cite{Hr1}, who applied it to the Kat\'etov order on almost disjoint families

On Completely Separable MAD Families (Set Theory : Reals and Topology)

We survey some constructions of completely separable MAD families

Crowded and Selective Ultrafilters under the Covering Property Axiom

In the paper we formulate an axiom CPA_{prism}^{game}, which is the most prominent version of the Covering Property Axiom CPA, and discuss several of its implications. In particular, we show that it implies that the following cardinal characteristics of continuum are equal to \omega1, while \continuum=\omega2: the independence number i, the reaping number r, the almost disjoint number a, and the ultrafilter base number u. We will also show that CPA_{prism}^{game} implies the existence of crowded and selective ultrafilters as well as nonselective P-points. In addition we prove that under CPA_{prism}^{game} every selective ultrafilter is \omega1-generated. The paper is finished with the proof that CPA_{prism}^{game} holds in the iterated perfect set model. It is known that the axiom CPA_{prism}^{game} captures the essence of the Sacks model concerning standard cardinal characteristics of continuum. This follows from a resent result of J. Zapletal who proved, assuming large cardinals, that for a ``nice\u27\u27 cardinal invariant \kappa if \kappa\u3c\continuum holds in any forcing extension than \kappa\u3c\continuum follows already from CPA_{prism}^{game}