On quasi-p-bounded subsets

The notion of quasi-p-boundedness for p ∈ ${\displaystyle {\omega }^{\cdot }}$ 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 ${\displaystyle {\omega }^{\cdot }}$ can be defined in terms of quasi-p-pseudocompactness. For p ∈ ${\displaystyle {\omega }^{\cdot }}$, we prove that a subset B of a space X is quasi-p-bounded in X if and only if B × ${\displaystyle P_{RK}\left(p\right)}$ is bounded in X × ${\displaystyle P_{RK}\left(p\right)}$, if and only if ${\displaystyle c{l}_{\beta (X\times P_{RK}\left(p\right))}(B\times P_{RK}\left(p\right))=c{l}_{\beta X}B\times \beta (\omega )}$, where ${\displaystyle P_{RK}\left(p\right)}$ is the set of Rudin-Keisler predecessors of p.