Biorthogonal systems and big quotient spaces

Finet, Catherine; Godefroy, Gilles

doi:10.1090/conm/085/983383

Cited by 23 publications

(34 citation statements)

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“…[18]). Given a metrizable compact M , its points {r ξ : ξ < κ} for some cardinal κ and the splitting continuous functions f ξ : M \ {r ξ } → K ξ where K ξ s are compact and metrizable we consider the split M induced by (f ξ ) ξ<κ , for precise definition see 2.1.…”

Section: That L Is Nonmetrizable and Totally Disconnected?mentioning

confidence: 99%

On the problem of compact totally disconnected reflection of nonmetrizability

Koszmider

2016

Topology and its Applications

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Abstract. We construct a ZFC example of a nonmetrizable compact space K such that every totally disconnected closed subspace L ⊆ K is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be of fixed big dimension. Then we focus on the problem if a nonmetrizable compact space K must have a closed subspace with a nonmetrizable totally disconnected continuous image. This question has several links with the the structure of the Banach space C(K), for example, by Holsztyński's theorem, if K is a counterexample, then C(K) contains no isometric copy of a nonseparable Banach space C(L) for L totally disconnected. We show that in the literature there are diverse consistent counterexamples, most eliminated by Martin's axiom and the negation of the continuum hypothesis, but some consistent with it. We analyze the above problem for a particular class of spaces. OCA+MA however, implies the nonexistence of any counterexample in this class but the existence of some other absolute example remains open.

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Section: That L Is Nonmetrizable and Totally Disconnected?mentioning

confidence: 99%

On the problem of compact totally disconnected reflection of nonmetrizability

Koszmider

2016

Topology and its Applications

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“…By the Sersouri result, we know that an uncountable ω-independent family F = {x α : α < ω 1 } always contains a convex right-separated family, so we may assume that F is convex right separated. By Lemma 2.1 (choosing a subfamily, if necessary) we can also suppose the existence of β 0 < ω 1 …”

Section: Theorem 22 Every Uncountable ω-Independent Family Fails Thmentioning

confidence: 99%

“…Later, assuming only the continuum hypothesis, Kunen [6] constructed a Banach space enjoying a stronger property: for every family {x α : α < ω 1 } there is α < ω 1 satisfying x α ∈ {x β : α < β} weak (recall that a family {x α : α < ω 1 } in a topological space is right separated if x α / ∈ {x β : α < β} for all α < ω 1 ). The Kunen space is an Asplund C(K) space with no Fréchet differentiable norms [4] and has many other interesting properties (see, for example, [1,3]). …”

Section: Introductionmentioning

confidence: 99%

ON $\omega$-INDEPENDENCE AND THE KUNEN–SHELAH PROPERTY

Granero¹,

Jiménez-Sevilla²,

Moreno³

2002

Proceedings of the Edinburgh Mathematical Society

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We prove that spaces with an uncountable ω-independent family fail the Kunen-Shelah property. Actually, if {x i } i∈I is an uncountable ω-independent family, there exists an uncountable subset J ⊂ I such that x j / ∈ conv({x i } i∈J\{j} ) for every j ∈ J. This improves a previous result due to Sersouri, namely that every uncountable ω-independent family contains a convex right-separated subfamily.

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“…It is asked in [FiG,Question IV.2] whether Kunen's C(Ω) space (see [N, pp. 1123-1129]) has a support set even though it has no uncountable biorthogonal system; note that Kunen's construction uses the continuum hypothesis.…”

Section: A Characterization Of Spaces Admitting Support Setsmentioning

confidence: 99%

“…For example one can check that the double arrow space Ω is hereditarily Lindelöf and hereditarily separable, while C(Ω) has an uncountable biorthogonal system; see e.g. [FiG,Example II.5(1)]. …”

Section: Introductionmentioning

confidence: 99%