Semimorasses and nonreflection at singular cardinals

Topology and its Applications

2016

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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.

Section: Introductionmentioning

confidence: 92%

On the problem of compact totally disconnected reflection of nonmetrizability

Topology and its Applications

2016

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“…As such spaces of singular sizes were already obtained in [22], our weakening of the hypothesis needed in this type of result is probably most interesting in case of λ = ω 2 as it is possible to construct (see Fact 36) a model of CH + KH ω 2 + ¬NR(ω 2 ) 2 and so it is consistent with CH that a nonreflecting nonmetrizable spaces exists but the classical example of Hajnal and Juhasz from the nonreflecting stationary set does not exists. It even follows from the Engelking-Lutzer and Balogh-Rudin characterizations of paracompactness in generalized ordered spaces and monotonically normal spaces (Fact 36), respectively, that It is consistent with CH that there is locally metrizable, first countable, nonreflecting nonmetrizable space of size ω 2 but there is no such space which is generalized ordered space or monotonically normal one.…”

Section: Introductionmentioning

confidence: 96%

“…The family F(T ) is a weak version of an (ω 1 , λ)-semimorass for λ = | Br(T )| (see [22]) which is a weak version of an (ω 1 , 1)-morass (see [33]), but it can also originate from the construction of a Kurepa tree from countable sets of its branches as is done in [32]. Already Todorcevic realized (private communication) that if T is a Kurepa tree "obtained" from (ω 1 , 1)-morass, then F(T ) is a simplified morass.…”

Section: Introductionmentioning

confidence: 99%

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Kurepa trees and topological non-reflection

Topology and its Applications

2005

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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.

“…The consistency of the existence of locally countable, well-founded, stationary coding sets for any cardinal λ of uncountable cofinality was proved e.g. in [Kosz3]; for more on the existence of stationary coding sets see [Sh3], [Ve]. At the end of this discussion let us note the following elementary Fact 9.10.…”

Section: Hereditary Characters Of Points In Compact Spacesmentioning

confidence: 99%

Forcing minimal extensions of Boolean algebras

Trans. Amer. Math. Soc.

1999

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Abstract. We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications: 1) We make Fedorchuk's method more flexible, obtaining, for every cardinal λ of uncountable cofinality, a consistent example of a Boolean algebra A λ whose every infinite homomorphic image is of cardinality λ and has a countable dense subalgebra (i.e., its Stone space is a compact S-space whose every infinite closed subspace has weight λ). In particular this construction shows that it is consistent that the minimal character of a nonprincipal ultrafilter in a homomorphic image of an algebra A can be strictly less than the minimal size of a homomorphic image of A, answering a question of J. D. Monk.2) We prove that for every cardinal of uncountable cofinality it is consistent that 2 ω = λ and both A λ and Aω 1 exist.3) By combining these algebras we obtain many examples that answer questions of J.D. Monk. 4) We prove the consistency of MA + ¬CH + there is a countably tight compact space without a point of countable character, complementing results of A. Dow, V. Malykhin, and I. Juhasz. Although the algebra of clopen sets of the above space has no ultrafilter which is countably generated, it is a subalgebra of an algebra all of whose ultrafilters are countably generated. This proves, answering a question of Arhangel skii, that it is consistent that there is a first countable compact space which has a continuous image without a point of countable character. 5) We prove that for any cardinal λ of uncountable cofinality it is consistent that there is a countably tight Boolean algebra A with a distinguished ultrafilter ∞ such that for every a ∞ the algebra A|a is countable and ∞ has hereditary character λ.