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A remark on a theorem of Solecki

Petr Holický, Luděk Zajíček, Miroslav Zelený (2005)

Commentationes Mathematicae Universitatis Carolinae

S. Solecki proved that if is a system of closed subsets of a complete separable metric space X , then each Suslin set S X which cannot be covered by countably many members of contains a G δ set which cannot be covered by countably many members of . We show that the assumption of separability of X cannot be removed from this theorem. On the other hand it can be removed under an extra assumption that the σ -ideal generated by is locally determined. Using Solecki’s arguments, our result can be used...

A semifilter approach to selection principles

Lubomyr Zdomsky (2005)

Commentationes Mathematicae Universitatis Carolinae

In this paper we develop the semifilter approach to the classical Menger and Hurewicz properties and show that the small cardinal 𝔤 is a lower bound of the additivity number of the σ -ideal generated by Menger subspaces of the Baire space, and under 𝔲 < 𝔤 every subset X of the real line with the property Split ( Λ , Λ ) is Hurewicz, and thus it is consistent with ZFC that the property Split ( Λ , Λ ) is preserved by unions of less than 𝔟 subsets of the real line.

A semifilter approach to selection principles II: τ * -covers

Lubomyr Zdomsky (2006)

Commentationes Mathematicae Universitatis Carolinae

Developing the idea of assigning to a large cover of a topological space a corresponding semifilter, we show that every Menger topological space has the property fin ( 𝒪 , T * ) provided ( 𝔲 < 𝔤 ) , and every space with the property fin ( 𝒪 , T * ) is Hurewicz provided ( Depth + ( [ ω ] 0 ) 𝔟 ) . Combining this with the results proven in cited literature, we settle all questions whether (it is consistent that) the properties P and Q [do not] coincide, where P and Q run over fin ( 𝒪 , Γ ) , fin ( 𝒪 , T ) , fin ( 𝒪 , T * ) , fin ( 𝒪 , Ω ) , and fin ( 𝒪 , 𝒪 ) .

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