Monotone normality

We show that a monotonically normal space X is paracompact if and only if for every increasing open cover {U α : α < κ} of X, there is a closed cover {Fnα : n < ω, α < κ} of X such that Fnα ⊂ Uα for n < ω, α < κ and PreliminariesThe B-property is introduced by P. Zenor in [10]. A space X is said to have the B-property if every increasing open cover {U α : α < κ} of X has an increasing open cover {V α : α < κ} of X such that V α ⊂ U α for each α < κ, where by {U α : α < κ} being increasing we mean U α ⊂ U β wherever α < β. It is well-known that paracompactness ⇒ B-property ⇒ countable paracompactness. But the implications are not reversible (cf. [5], [6], [8]).

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“…Lemma 1 (Corollary 2.1 (d) of [1]). A monotonically normal space with the B-property is paracompact.…”

Section: Resultsmentioning

confidence: 97%

A note on monotonically normal spaces

Gao

Wang

2007

Acta Math Hung

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“…Observe that the proof of the theorem uses only one property of L, namely, the fact that L contains a closed copy of a stationary subset of an uncountable regular cardinal τ . Since the theorem of Engelking and Lutzer holds for monotonically normal spaces as well (proved by Balogh and Rudin [1]) we have the following. Theorem 1.4.…”

Section: Theorem (R Engelking and D Lutzer [3])mentioning

confidence: 99%

On an algebraic version of Tamano’s theorem

Buzyakova

2009

Appl.Gen.Topol.

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“…By the assumption, ∆ i θ is nonempty for 1 i k θ and θ in some stationary set S . By thinning out S , find stationary S 0 ⊆ S ∩ Lim(ω 1 ), k ∈ ω such that for every θ 1 , θ 2 ∈ S 0 we have k θ 1 …”

Section: Lemma 18 Letmentioning

confidence: 99%

“…First let us prove the cofinality of F(T ) in [Br(T )] ω . Let X ∈ [Br(T )] ω , find θ < ω 1 such that for every b 1 …”

Section: F (T ) and The Ideal Of Light Setsmentioning

confidence: 99%

“…1 A cardinal κ is called singular if sets of cardinality κ can be presented as α<λ A α where the cardinality of each A α is less than κ as well as λ < κ. Thus, when a property P does not reflect at a singular cardinal, it means in particular that there is a structure S with property P which is a union of less than |S| substructures without property P .…”

Section: Introductionmentioning

confidence: 99%

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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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Monotone normality

Cited by 72 publications

References 2 publications

A note on monotonically normal spaces

A note on monotonically normal spaces

On an algebraic version of Tamano’s theorem

Kurepa trees and topological non-reflection

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