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Covering properties in countable products, II

Sachio Higuchi, Hidenori Tanaka (2006)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we discuss covering properties in countable products of Čech-scattered spaces and prove the following: (1) If Y is a perfect subparacompact space and { X n : n ∈ ω } is a countable collection of subparacompact Čech-scattered spaces, then the product Y × ∏ n ∈ ω X n is subparacompact and (2) If { X n : n ∈ ω } is a countable collection of metacompact Čech-scattered spaces, then the product ∏ n ∈ ω X n is metacompact.

Covering Σ ξ 0 -generated ideals by Π ξ 0 sets

Tamás Mátrai (2007)

Commentationes Mathematicae Universitatis Carolinae

We develop the theory of topological Hurewicz test pairs: a concept which allows us to distinguish the classes of the Borel hierarchy by Baire category in a suitable topology. As an application we show that for every Π ξ 0 and not Σ ξ 0 subset P of a Polish space X there is a σ -ideal ℐ ⊆ 2 X such that P ∉ ℐ but for every Σ ξ 0 set B ⊆ P there is a Π ξ 0 set B ' ⊆ P satisfying B ⊆ B ' ∈ ℐ . We also discuss several other results and problems related to ideal generation and Hurewicz test pairs.

Covering ω ω by special Cantor sets

Gary Gruenhage, Ronnie Levy (2002)

Commentationes Mathematicae Universitatis Carolinae

This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ω ω of irrationals, or certain of its subspaces. In particular, given f ∈ ω ( ω ∖ { 0 } ) , we consider compact sets of the form ∏ i ∈ ω B i , where | B i | = f ( i ) for all, or for infinitely many, i . We also consider “ n -splitting” compact sets, i.e., compact sets K such that for any f ∈ K and i ∈ ω , | { g ( i ) : g ∈ K , g ↾ i = f ↾ i } | = n .

Criterion of Normality of the Completely Regular Topology of Separate Continuity

Grinshpon, Yakov S. (2006)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 54C10, 54D15, 54G12.For given completely regular topological spaces X and Y, there is a completely regular space X ~⊗ Y such that for any completely regular space Z a mapping f : X × Y ⊗ Z is separately continuous if and only if f : X ~⊗ Y→ Z is continuous. We prove a necessary condition of normality, a sufficient condition of collectionwise normality, and a criterion of normality of the products X ~⊗ Y in the case when at least one factor is scattered.

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