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On quasi-p-bounded subsets

100%

Sanchis M. , Tamariz-Mascarúa A.

Colloquium Mathematicum

1999

tom 80

nr 2

175-189

The notion of quasi-p-boundedness for p ∈ $ω^*$ is introduced and investigated. We characterize quasi-p-pseudocompact subsets of β(ω) containing ω, and we show that the concepts of RK-compatible ultrafilter and P-point in $ω^*$ can be defined in terms of quasi-p-pseudocompactness. For p ∈ $ω^*$, we prove that a subset B of a space X is quasi-p-bounded in X if and only if B × $P_{RK}(p)$ is bounded in X × $P_{RK}(p)$, if and only if $cl_{β(X × P_{RK}(p))}(B× P_{RK}(p)) = cl_{βX} B × β(ω)$, where $P_{RK}(p)$ is the set of Rudin-Keisler predecessors of p.

Crowded and selective ultrafilters under the covering property axiom

59%

Ciesielski K. , Pawlikowski J.

Journal of Applied Analysis

2003

tom Vol. 9, nr 1

19-55

In the paper we formulate an axiom CPAgame prism, 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 ω 1 , while c = ω 2 : the independence number i, the reaping number r, the almost disjoint number a, and the ultrafilter base number u. We will also show that CPAgame prism, implies the existence of crowded and selective ultrafilters as well as nonselective P-points. In addition we prove that under CPAgame prism every selective ultrafilter is ω 1-generated. The paper finishes with the proof that CPAgame prism holds in the iterated perfect set model.