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The partially pre-ordered set of compactifications of Cp(X, Y)

A. Dorantes-Aldama, R. Rojas-Hernández, Á. Tamariz-Mascarúa (2015)

Topological Algebra and its Applications

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...

The Rothberger property on C p ( Ψ ( 𝒜 ) , 2 )

Daniel Bernal-Santos (2016)

Commentationes Mathematicae Universitatis Carolinae

A space X is said to have the Rothberger property (or simply X is Rothberger) if for every sequence 𝒰 n : n ω of open covers of X , there exists U n 𝒰 n for each n ω such that X = n ω U n . For any n ω , necessary and sufficient conditions are obtained for C p ( Ψ ( 𝒜 ) , 2 ) n to have the Rothberger property when 𝒜 is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family 𝒜 for which the space C p ( Ψ ( 𝒜 ) , 2 ) n is Rothberger for all n ω .

The subspace of weak P -points of *

Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)

Commentationes Mathematicae Universitatis Carolinae

Let W be the subspace of * consisting of all weak P -points. It is not hard to see that W is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that W is a p -pseudocompact space for all p * .

The σ -property in C ( X )

Anthony W. Hager (2016)

Commentationes Mathematicae Universitatis Carolinae

The σ -property of a Riesz space (real vector lattice) B is: For each sequence { b n } of positive elements of B , there is a sequence { λ n } of positive reals, and b B , with λ n b n b for each n . This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “ σ ” obtains for a Riesz space of continuous real-valued functions C ( X ) . A basic result is: For discrete X , C ( X ) has σ iff the cardinal | X | < 𝔟 , Rothberger’s bounding number. Consequences and...

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