Critical Cardinalities and Additivity Properties of Combinatorial Notions of Smallness

Shelah, Saharon; Tsaban, Boaz

doi:10.1515/jaa.2003.149

Cited by 11 publications

(20 citation statements)

References 13 publications

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“…Solving a longstanding problem, Malliaris and the second named author have recently proved that p = t [6]. We prove that, consistently, p < lr < c. This settles [16, Problem 64] (quoted in [15,Problem 5] and in [17,Problem 11.2 (311)]). Moreover, we have that lr = d in all models of set theory where the continuum is at most ℵ 2 .…”

Section: Overviewsupporting

confidence: 65%

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The linear refinement number and selection theory

2016

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The \emph{linear refinement number} $\mathfrak{lr}$ is the minimal cardinality of a centered family in $[\omega]^\omega$ such that no linearly ordered set in $([\omega]^\omega,\subseteq^*)$ refines this family. The \emph{linear excluded middle number} $\mathfrak{lx}$ is a variation of $\mathfrak{lr}$. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classic combinatorial cardinal characteristics of the continuum. We prove that $\mathfrak{lr}=\mathfrak{lx}=\mathfrak{fd}$ in all models where the continuum is at most $\aleph_2$, and that the cofinality of $\mathfrak{lr}$ is uncountable. Using the method of forcing, we show that $\mathfrak{lr}$ and $\mathfrak{lx}$ are not provably equal to $\mathfrak{d}$, and rule out several potential bounds on these numbers. Our results solve a number of open problems

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Section: Overviewsupporting

confidence: 65%

“…It is known that lr ≤ lx ≤ d [16] and that b, s ≤ lx [15]. In particular, by the abovementioned result on lr, we have that lx = d whenever the continuum is at most ℵ 2 .…”

Section: Overviewmentioning

confidence: 72%

The linear refinement number and selection theory

2016

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“…add(X, D) is the minimal cardinality κ of a dominating Y ⊆ N N such that for each partition of Y into κ many pieces, there is a piece such that no g avoids middles in that piece. This cardinal is studied in [59]. Problem 11.5 ([59]).…”

Section: Cardinal Characteristics Of the Continuummentioning

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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“…An element U ∈ O(X) is in T(X) if every member of X is a member of infinitely many elements of U, and, for all x, y ∈ X, either x ∈ U implies y ∈ U for all but finitely many U ∈ U, or y ∈ U implies x ∈ U for all but finitely many U ∈ U. Figure 2 contains all new properties introduced by the inclusion of T into the framework, together with their critical cardinalities [42,39,21,20], and a serial number to be used below. Proof.…”

Section: Applicationsmentioning

confidence: 99%

Selective covering properties of product spaces, II: $\gamma $ spaces

Miller

Tsaban

Zdomskyy

2015

Trans. Amer. Math. Soc.

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Abstract. We study productive properties of γ spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results:(1) Solving a problem of F. Jordan, we show that for every unbounded tower set X ⊆ R of cardinality ℵ 1 , the space C p (X) is productively Fréchet-Urysohn. In particular, the set X is productively γ. (2) Solving problems of Scheepers and Weiss, and proving a conjecture of BabinkostovaScheepers, we prove that, assuming the Continuum Hypothesis, there are γ spaces whose product is not even Menger. (3) Solving a problem of Scheepers-Tall, we show that the properties γ and Gerlits-Nagy (*) are preserved by Cohen forcing. Moreover, every Hurewicz space that Remains Hurewicz in a Cohen extension must be Rothberger (and thus (*)). We apply our results to solve a large number of additional problems, and use Arhangel'skiȋ duality to obtain results concerning local properties of function spaces and countable topological groups.

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Critical Cardinalities and Additivity Properties of Combinatorial Notions of Smallness

Cited by 11 publications

References 13 publications

The linear refinement number and selection theory

The linear refinement number and selection theory

Selection principles and special sets of reals

Selective covering properties of product spaces, II: $\gamma $ spaces

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