New proofs of the consistency of the normal Moore space conjecture II

Dow, Alan; Tall, Franklin D.; Weiss, William

doi:10.1016/0166-8641(90)90056-8

Cited by 26 publications

(27 citation statements)

References 12 publications

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“…If the nice subspace that we seek means a subspace of small cardinality, positive consistent answers to this question were obtained by Alan Dow and others, for example, in [8], [10], [11], [24], [46]. When one restrict oneself to compact Hausdorff spaces the question if every nonmetrizable compact Hausdorff space has a nonmetrizable subspace of cardinality ω 1 has the positive answer in ZFC as proved by Alan Dow in [9].…”

Section: Introductionmentioning

confidence: 92%

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: Introductionmentioning

confidence: 92%

On the problem of compact totally disconnected reflection of nonmetrizability

Koszmider

2016

Topology and its Applications

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“…The incompactness (i.e., our non-reflection) of second order properties is highly nontrivial and has been extensively studied in combinatorics (e.g. [28]), group theory (for a references see [26]) or topology itself [6,7,10,18,30].…”

Section: Introductionmentioning

confidence: 99%

Kurepa trees and topological non-reflection

Koszmider

2005

Topology and its Applications

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A property P of a structure S does not reflect if no substructure of S of smaller cardinality than S has the property. If for a given property P there is such an S of cardinality κ, we say that P does not reflect at κ. We undertake a fine analysis of Kurepa trees which results in defining canonical topological and combinatorial structures associated with the tree which possess a remarkably wide range of nonreflecting properties providing new constructions and solutions of open problems in topology. The most interesting results show that many known properties may not reflect at any fixed singular cardinal of uncountable cofinality. The topological properties we consider vary from normality, collectionwise Hausdorff property to metrizablity and many others. The combinatorial properties are related to stationary reflection.

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“…The Cohen real case was proved by Dow, Tall, and Weiss [DTW]. Corollary 1.3 [Nl' F 5 ,DTW]. In the models V [G] o/Theorem 1.1, every normal space 0/ character < 2(j) is collection wise normal.…”

Section: The Normal Locally Compact Space Hypothesis (Nlch) Every Nomentioning

confidence: 99%

On collectionwise normality of locally compact, normal spaces

Balogh¹

1991

Trans. Amer. Math. Soc.

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ABSTRACT. We prove that by adjoining supercompact many Cohen or random reals to a model of ZFC set theory, in the resulting model, every normal locally compact space is collectionwise normal. In the same models, countably paracompact, locally compact T 3 -spaces are expandable. Local compactness in the above theorems can be weakened to being of point-countable type, a condition that is implied by both tech-completeness and first countability.

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New proofs of the consistency of the normal Moore space conjecture II

Cited by 26 publications

References 12 publications

On the problem of compact totally disconnected reflection of nonmetrizability

On the problem of compact totally disconnected reflection of nonmetrizability

Kurepa trees and topological non-reflection

On collectionwise normality of locally compact, normal spaces

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