On Banach spaces whose norm-open sets are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>F</mml:mi><mml:mi>σ</mml:mi></mml:msub></mml:math>-sets in the weak topology

Marciszewski, Witold; Pol, Roman

doi:10.1016/j.jmaa.2008.06.007

Cited by 21 publications

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“…Using the arguments of Section 5 of [2] (cf. [35]) one can see that there are pairwise nonisomorphic spaces satisfying Theorem 1. It should be clear that the almost disjoint family of branches of the Cantor tree is -embeddable and such families are Borel in the product topology on (see, e.g., Lemma 30 of [28]), so it follows from the results of [36] that the spaces satisfying Theorem 1 can be isomorphically of the form , where is a Rosenthal compact space.…”

Section: Introductionmentioning

confidence: 99%

Banach Spaces in Which Large Subsets of Spheres Concentrate

Koszmider

2022

J. Inst. Math. Jussieu

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We construct a nonseparable Banach space $\mathcal {X}$ (actually, of density continuum) such that any uncountable subset $\mathcal {Y}$ of the unit sphere of $\mathcal {X}$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon $ for some $\varepsilon>0$ ). This solves in the negative the central problem of the search for a nonseparable version of Kottman’s theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains neither an uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson–Lindenstrauss space induced by an $\mathbb {R}$ -embeddable almost disjoint family of subsets of $\mathbb {N}$ . We also show that this special feature of the almost disjoint family is essential to obtain the above properties.

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Section: Introductionmentioning

confidence: 99%

Banach Spaces in Which Large Subsets of Spheres Concentrate

Koszmider

2022

J. Inst. Math. Jussieu

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“…Almost disjoint families, especially MADF's, are of interest in set theory (e.g., [8], [14]), topology (e.g., [6], [13]), Boolean algebra (e.g., [1], [3]) and Banach spaces (e.g., [10], [11]).…”

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confidence: 99%

Ordinal remainders of classical ψ-spaces

Dow

Vaughan

2012

Fund. Math.

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Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain {Tα : α < λ} of infinite subsets of ω, there exists M ⊂ [ω] ω , an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ \ ψ of the associated ψ-space, ψ = ψ(ω, M), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < t + , where t is the tower number, there exists a mod-finite ascending chain {Tα : α < λ}, hence a ψ-space with Stone-Čech remainder homeomorphic to λ + 1. This generalizes a result credited to S. Mrówka by J. Terasawa which states that there is a MADF M such that βψ \ ψ is homeomorphic to ω1 + 1.

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Separably Injective Banach Spaces

Avilés

Cabello

Castillo

et al. 2016

Lecture Notes in Mathematics

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Abstract. In this paper we deal with two weaker forms of injectivity which turn out to have a rich structure behind: separable injectivity and universal separable injectivity. We show several structural and stability properties of these classes of Banach spaces. We provide natural examples of (universally) separably injective spaces, including L∞ ultraproducts built over countably incomplete ultrafilters, in spite of the fact that these ultraproducts are never injective. We obtain two fundamental characterizations of universally separably injective spaces: a) A Banach space E is universally separably injective if and only if every separable subspace is contained in a copy of ℓ∞ inside E. b) A Banach space E is universally separably injective if and only if for every separable space S one has Ext(ℓ∞/S, E) = 0. The final Section of the paper focuses on special properties of 1-separably injective spaces. Lindenstrauss obtained in the middle sixties a result that can be understood as a proof that, under the continuum hypothesis, 1-separably injective spaces are 1-universally separably injective; he left open the question in ZFC. We construct a consistent example of a Banach space of type C(K) which is 1-separably injective but not 1-universally separably injective.

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On Banach spaces whose norm-open sets are Fσ-sets in the weak topology

Cited by 21 publications

References 16 publications

Banach Spaces in Which Large Subsets of Spheres Concentrate

Banach Spaces in Which Large Subsets of Spheres Concentrate

Ordinal remainders of classical ψ-spaces

Separably Injective Banach Spaces

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