Countable paracompactness versus normality in ω21

Kemoto, Nobuyuki; Smith, Kerry D.; Szeptycki, Paul J.

doi:10.1016/s0166-8641(99)00023-1

Cited by 8 publications

(3 citation statements)

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“…In any model of ZFC where first countable countably paracompact spaces are strongly collectionwise Hausdorff, we have a positive answer to Question 3. Although it is an open question (due to P. Nyikos) whether V = L implies that countably paracompact first countable spaces are strongly collectionwise Hausdorff, it was shown in [9] that Question 3 still has a positive answer assuming V = L.…”

Section: Consistency Of Thin Ladder Systemsmentioning

confidence: 99%

Uniformization and anti-uniformization properties of ladder systems

et al. 2004

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Abstract. Natural weakenings of uniformizability of a ladder system on ω 1 are considered. It is shown that even assuming CH all the properties may be distinct in a strong sense. In addition, these properties are studied in conjunction with other properties inconsistent with full uniformizability, which we call anti-uniformization properties. The most important conjunction considered is the uniformization property we call countable metacompactness and the anti-uniformization property we call thinness. The existence of a thin, countably metacompact ladder system is used to construct interesting topological spaces: a countably paracompact and nonnormal subspace of ω 2 1 , and a countably paracompact, locally compact screenable space which is not paracompact. Whether the existence of a thin, countably metacompact ladder system is consistent is left open. Finally, the relation between the properties introduced and other well known properties of ladder systems, such as ♣, is considered.

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Section: Consistency Of Thin Ladder Systemsmentioning

confidence: 99%

Uniformization and anti-uniformization properties of ladder systems

et al. 2004

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“…This space X is a specific instance of a class of nonnormal spaces (implicit in Lemma 4 of [4], and Lemma 5.1.3 of [5]). Let κ be an uncountable regular cardinal.…”

Section: Definition 32 For Sets S and T Let The Diagonal Map Dgmentioning

confidence: 99%

Normal subspaces of products of finitely many ordinals

Fleissner

2002

Proc. Amer. Math. Soc.

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Abstract. Let X be a subspace of the product of finitely many ordinals. If X is normal, then X is strongly zero-dimensional, collectionwise normal, and shrinking. The proof uses (κ 1 , . . . , κn)-stationary sets. PreliminaryWe use (κ 1 , . . . , κ n )-stationary sets to prove the following theorem which extends results of Kemoto, Nogura, Smith, and Yajima in [4] and Stanley in [8]. (1) X is normal.(2) X is normal and strongly zero-dimensional.This theorem differs from the theorem of [4] in two ways. First, that theorem applies to subspaces of the product of two ordinals. Second, that theorem does not include "strongly zero-dimensional". Instead, the fourth condition asserts that nine types of pairs of closed sets are separated. For example, if (µ, ν) / ∈ X ⊆ λ 2 , then {(α, ν) ∈ X : α < λ} and {(µ, β) ∈ X : β < λ} are separated. Stanley's theorem asserts that if X is a normal subspace of the product of finitely many ordinals, then X is collectionwise Hausdorff. The condition that X is strongly zero-dimensional cannot be added to Theorem 1.1 because the paper [3] describes a subspace of ω + 1 × c which is not strongly zero-dimensional.First we define the notions in Theorem 1.1, and then introduce notation for tuples and products.

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“…La prueba del siguiente teorema la omitiremos, ya que está fuera del propósito de este trabajo. Se puede consultar una demostración en [13]. 1 es estrella Lindelöf si y sólo si tiene extensión numerable.…”

Section: Supongamos Que Se Cumple (1) Comounclassified

Propiedades topológicas del tipo estrella P

Domínguez

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Definición 1.1.4. Un espacio de Tychonoff X es pseudocompacto si para cada f ∈ C(X), se tiene que f [X] es acotado en R, es decir, C(X) = C * (X).Proposición 1.1.5. Si X es un espacio de Tychonoff numerablemente compacto, entonces X es pseudocompacto.Demostración. Sea f → R una función continua, entonces f [X] es numerablemente compacto en R. Como ser compacto y numerablemente compacto son propiedades equivalentes en espacios métricos, tenemos que f [X] compacto, por lo tanto f [X] es acotado.El siguiente lema es el Corolario 3.10.27 de [6].Lema 1.1.6. Si X es un espacio pseudocompacto y Y es un espacio compacto, entonces X × Y es pseudocompacto.Lema 1.1.7. Sea X un espacio topológico de Tychonoff y D ⊆ X un subespacio denso y pseudocompacto, entonces X es pseudocompacto. DemostraciónPor lo tanto X es pseudocompacto.Ejemplo 1.1.8. Existe un espacio que es pseudocompacto y que no es numerablemente compacto.

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Countable paracompactness versus normality in ω21

Cited by 8 publications

References 0 publications

Uniformization and anti-uniformization properties of ladder systems

Uniformization and anti-uniformization properties of ladder systems

Normal subspaces of products of finitely many ordinals

Propiedades topológicas del tipo estrella P

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