Generalized paracompactness of subspaces in products of two ordinals

Kemoto, Nobuyuki; Tamano, Kenichi; Yajima, Yukinobu

doi:10.1016/s0166-8641(99)00010-3

Cited by 6 publications

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“…. , 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.…”

supporting

confidence: 52%

“…This theorem differs from the theorem of [4] in two ways. First, that theorem applies to subspaces of the product of two ordinals.…”

Section: Theorem 11 Let X Be a Subspace Of The Product Of Finitely mentioning

confidence: 80%

“…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%

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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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“…. , 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.…”

supporting

confidence: 52%

“…This theorem differs from the theorem of [4] in two ways. First, that theorem applies to subspaces of the product of two ordinals.…”

Section: Theorem 11 Let X Be a Subspace Of The Product Of Finitely mentioning

confidence: 80%

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Normal subspaces of products of finitely many ordinals

Fleissner

2002

Proc. Amer. Math. Soc.

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“…Furthermore a 0-unbounded sequence is said to be a 0-normal sequence for x if x(α) = sup{x(β): β < α} for each limit α < κ. If μ is an ordinal and cf μ ω and M : cf μ → μ is a function such that {M(α): α ∈ cf μ} is a normal sequence of μ, then the function M is said to be a normal function, where {M(α): α ∈ cf μ} is said to be normal if M(α) = sup{M(β): β < α} for each limit α < cf μ and μ = sup{M(α): α ∈ cf μ} (see [12] and [14]). …”

Section: Resultsmentioning

confidence: 99%

Products of certain dually discrete spaces

Peng

2009

Topology and its Applications

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“…Kemoto, Tamano, and Yajima ( [9]) proved that metacompactness, screenability, and weak submetalindelöfness are equivalent for subspaces of the product of two ordinals.…”

Section: Has No Closed Subset Homeomorphic To a Stationary Subset Omentioning

confidence: 99%

Metacompact subspaces of products of ordinals

Fleissner

2001

Proc. Amer. Math. Soc.

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Abstract. Let X be a subspace of the product of finitely many ordinals. X is countably metacompact, and X is metacompact iff X has no closed subset homeomorphic to a stationary subset of a regular uncountable cardinal. A theorem generalizing these two results is: X is λ-metacompact iff X has no closed subset homeomorphic to a (κ 1 , . . . , κn)-stationary set where κ 1 < λ. SourcesThis paper combines two lines of research. The first is the investigation of countably metacompact subspaces of the product of ordinals by Kemoto and Smith in [7] and [8]. A synthesis of the main theorems from these papers is proved that a subspace of a linearly ordered space is a D-space iff it is metacompact iff it has no closed subset homeomorphic to a stationary subset of a regular, uncountable cardinal. Stanley ([11] and [3]) proved the same equivalence for subspaces of the product of finitely many ordinals. Theorem 1.2. Let X be a subspace of the product of finitely many ordinals. The following are equivalent: X has no closed subset homeomorphic to a stationary subset of a regular uncountable cardinal.Kemoto, Tamano, and Yajima ([9]) proved that metacompactness, screenability, and weak submetalindelöfness are equivalent for subspaces of the product of two ordinals.Considering these results, it is natural to conjecture first, that every subspace of a finite product of ordinals is countably metacompact, and second, there is a

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Generalized paracompactness of subspaces in products of two ordinals

Cited by 6 publications

References 0 publications

Normal subspaces of products of finitely many ordinals

Normal subspaces of products of finitely many ordinals

Products of certain dually discrete spaces

Metacompact subspaces of products of ordinals

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