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

In the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where...