On the problem of compact totally disconnected reflection of nonmetrizability

Koszmider, Piotr

doi:10.1016/j.topol.2016.08.017

Cited by 6 publications

(3 citation statements)

References 33 publications

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“…There are known several constructions, using some additional set-theoretic assumptions, of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. Recently, Koszmider [9] constructed the first such example without such additional assumptions, and Plebanek [15] constructed a consistent example of such a space which is a Corson compact. So it is important to determine whether we can obtain such examples within some other known classes of compact spaces, for example Eberlein compact spaces.…”

Section: On Zero-dimensional Closed Subspaces Of Nonmetrizable Eberlein Compactamentioning

confidence: 99%

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On two problems concerning Eberlein compacta

Marciszewski

2021

RACSAM

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We discuss two problems concerning the class Eberlein compacta, i.e., weakly compact subspaces of Banach spaces. The first one deals with preservation of some classes of scattered Eberlein compacta under continuous images. The second one concerns the known problem of the existence of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. We show that the existence of such Eberlein compacta is consistent with . We also show that it is consistent with that each Eberlein compact space of weight $$> \omega _1$$ > ω 1 contains a nonmetrizable closed zero-dimensional subspace.

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Section: On Zero-dimensional Closed Subspaces Of Nonmetrizable Eberlein Compactamentioning

confidence: 99%

“…Several such spaces were obtained using some additional set-theoretic assumptions. Recently, Koszmider [9] constructed the first such example in ZFC. We investigate this problem for the class of Eberlein compact spaces.…”

Section: Introductionmentioning

confidence: 99%

On two problems concerning Eberlein compacta

Marciszewski

2021

RACSAM

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“…Note that Theorem 3.4 says in particular that the space K in question contains no nonmetrizable zero-dimensional subspaces. Recall that such a feature is somewhat delicate; only quite recently Koszmider [17] constructed a ZFC example of a nonmetrizable compact space having that property. Assuming the existence of a Lusin set, Marciszewski [21] gives another example of that kind which is an Eberlein compact space.…”

Section: Claim the Set πmentioning

confidence: 99%

Musing on Kunen's compact $L$-space

Plebanek¹

2020

Preprint

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Uncountable equilateral sets in Banach spaces of the form C(K)

Koszmider

2018

Isr. J. Math.

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The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin's axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form C(K) has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form C(K) which has no uncountable equilateral set (or equivalently no uncountable (1 + ε)separated set in the unit sphere for any ε > 0) making another consistent combinatorial assumption. The compact K is a version of the split interval obtained from a sequence of functions which behave in an anti-Ramsey manner. It remains open if there is an absolute example of a nonseparable Banach space of the form different than C(K) which has no uncountable equilateral set. It follows from the results of S. Mercourakis, G. Vassiliadis that our example has an equivalent renorming in which it has an uncountable equilateral set. It remains open if there are consistent examples which have no uncountable equilateral sets in any equivalent renorming but it follows from the results of S. Todorcevic that it is consistent that every nonseparable Banach space has an equivalent renorming in which it has an uncountable equilateral set.

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On the problem of compact totally disconnected reflection of nonmetrizability

Cited by 6 publications

References 33 publications

On two problems concerning Eberlein compacta

On two problems concerning Eberlein compacta

Musing on Kunen's compact $L$-space

Uncountable equilateral sets in Banach spaces of the form C(K)

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