Continuous images of sets of reals

Bartoszyński, Tomek; Shelah, Saharon

doi:10.1016/s0166-8641(00)00079-1

Cited by 21 publications

(21 citation statements)

References 9 publications

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“…Let g ∈ N N be a witness for that. By induction on n, choose finite subsets F n ⊆ U n such that m≤g(n) C n m ⊆ F n , and such that F n is not equal to any F k for k < n. 3 Consequently,…”

Section: Connections With Selection Principlesmentioning

confidence: 99%

Scales, fields, and a problem of Hurewicz

Tsaban¹,

Zdomskyy²

2008

J. Eur. Math. Soc.

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Menger's basis property is a generalization of σ-compactness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non σcompact sets of reals with Menger's property. Special instances of these constructions give known counterexamples to conjectures of Menger and Hurewicz. We obtain the first explicit solution to the Hurewicz 1927 problem, that was previously solved by Chaber and Pol on a dichotomic basis.The constructed sets generate nontrivial subfields of the real line with strong combinatorial properties, and most of our results can be stated in a Ramsey-theoretic manner.Since we believe that this paper is of interest to a diverse mathematical audience, we have made a special effort to make it selfcontained and accessible.Whenever you can settle a question by explicit construction, be not satisfied with purely existential arguments.

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Section: Connections With Selection Principlesmentioning

confidence: 99%

Scales, fields, and a problem of Hurewicz

Tsaban¹,

Zdomskyy²

2008

J. Eur. Math. Soc.

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“…The answer becomes "Yes" if we are allowed to pick, instead of one element from each cover, a union of two elements from each cover [71]. There is a direct construction of a set of reals H satisfying U fin (O, Γ) such that |H| = b (and such that H does not contain a perfect set) [10]. All finite powers of this set H satisfy U fin (O, Γ) [12].…”

Section: Examples Without Special Set Theoretic Hypothesesmentioning

confidence: 99%

Selection principles and special sets of reals

Tsaban¹

2007

Open Problems in Topology II

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We give a selection of major open problems involving selective properties, diagonalizations, and covering properties for sets of real numbers. This is a revision of the version published as a chapter in the book Open Problems in Topology II (E. Pearl, ed.), Elsevier B.V., 2007, 91-108. The present version reports solutions of some problems, uses up-to-date notation, and has updated bibliography.Comments and further updates would be appreciated.

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“…A set of reals is totally imperfect if it has no perfect subsets. There are in the literature several examples of totally imperfect sets of reals satisfying U f in (O, Γ), see [3,1,5]. However, all of them actually satisfy S 1 (Γ, Γ) [4] or at least S 1 (Γ, O) [2,5].…”

Section: Problem Of the Issuementioning

confidence: 99%