Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures

Self Cite

Abstract. We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of R (thus strictly o-bounded) which have the Menger and Hurewicz properties but are not σ-compact, and show that the product of two o-bounded subgroups of R N may fail to be o-bounded, even when they satisfy the stronger property S 1 (B Ω , B Ω ). This solves a problem of Tkačenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups G of size continuum such that every countable Borel ω-cover of G contains a γ-cover of G.

Section: Two Almost σ-Compact Subgroups Of Rmentioning

confidence: 99%

Section: Lemma 22 S 1 (B ω B γ ) Is Preserved Under Taking Countabmentioning

confidence: 99%

𝑜-bounded groups and other topological groups with strong combinatorial properties

Tsaban

2005

Self Cite

“…Our construction is essentially due to [4,Theorem 16]. Let D = {f α : α < d} be a dominating subset of ω ω which satisfies the condition (*): for…”

Section: Corollary 36 If X Is a Lusin Set Or A Sierpiński Set Thenmentioning

confidence: 99%

Menger subsets of the Sorgenfrey line

Sakai

2009

Abstract. A space X is said to have the Menger property if for every sequence {U n : n ∈ ω} of open covers of X, there are finite subfamilies V n ⊂ U n (n ∈ ω) such that n∈ω V n is a cover of X. Let i : S → R be the identity map from the Sorgenfrey line onto the real line and let X S = i −1 (X) for X ⊂ R. Lelek noted in 1964 that for every Lusin set L in R, L S has the Menger property. In this paper we further investigate Menger subsets of the Sorgenfrey line. Among other things, we show: (1) If X S has the Menger property, then X has Marczewski's property (s 0 ). (2) Let X be a zero-dimensional separable metric space. If X has a countable subset Q satisfying that X \ A has the Menger property for every countable set A ⊂ X \ Q, then there is an embedding e : X → R such that e(X) S has the Menger property. (3) For a Lindelöf subspace of a real GO-space (for instance the Sorgenfrey line), total paracompactness, total metacompactness and the Menger property are equivalent.

“…Every σ-compact space satisfies U fin (O, Γ), but the converse fails [7,2]. Let A and B be any two families.…”

Section: Introductionmentioning

confidence: 99%

Hurewicz sets of reals without perfect subsets

Repovš¹,

Tsaban

Zdomskyy

2008

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Abstract. We show that even for subsets X of the real line that do not contain perfect sets, the Hurewicz property does not imply the property S 1 (Γ, Γ), asserting that for each countable family of open γ-covers of X, there is a choice function whose image is a γ-cover of X. This settles a problem of Just, Miller, Scheepers, and Szeptycki. Our main result also answers a question of Bartoszyński and the second author, and implies that for C p (X), the conjunction of Sakai's strong countable fan tightness and the Reznichenko property does not imply Arhangel skiȋ's property α 2 .