κ-normality and products of ordinals

Kalantan, Lutfi; Szeptycki, Paul J.

doi:10.1016/s0166-8641(01)00220-6

Cited by 16 publications

(19 citation statements)

References 9 publications

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“…Nagami [17] proved that if every finite subproduct of X = i∈ω X i has covering dimension n and X is normal, then X has also covering dimension n. With the result of [8], these results yield: Lemma 1.9 [10,17,8]. Let α i be an ordinal and…”

Section: Strong Zero-dimensionality and κ -Normalitymentioning

confidence: 96%

“…Note that the proof does not use elementary submodel techniques. Lemma 1.8 [10]. Let α i be an ordinal and Y i = {β < α i : cf β ω} for each i ∈ ω.…”

Section: Strong Zero-dimensionality and κ -Normalitymentioning

confidence: 99%

“…In a different line Kalantan and Szeptycki [10] proved that arbitrary products of ordinals are κ-normal: every disjoint pair of regular closed sets is separated by disjoint open sets. In contrast, if A, B and C are stationary subsets of ω 1 such that any two have stationary intersection but the intersection of all three is not stationary, then the product A × B × C is not κ-normal [9].…”

Section: Introductionmentioning

confidence: 99%

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Topological properties of products of ordinals

Kemoto

Szeptycki

2004

Topology and its Applications

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We study separation and covering properties of special subspaces of products of ordinals. In particular, it is proven that certain subspaces of Σ-products of ordinals are quasi-perfect preimages of Σ-products of copies of ω. We obtain as corollaries that products of ordinals are κ-normal and strongly zero-dimensional. Also, σ -products and Σ-products of ordinals are shown to be countably paracompact, κ-normal and strongly zero-dimensional. Normality in Σ-products and σ -products of ordinals is also characterized. It is also shown that any continuous real-valued function on a σ -product of ordinals has countable range.

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Section: Strong Zero-dimensionality and κ -Normalitymentioning

confidence: 96%

“…Note that the proof does not use elementary submodel techniques. Lemma 1.8 [10]. Let α i be an ordinal and Y i = {β < α i : cf β ω} for each i ∈ ω.…”

Section: Strong Zero-dimensionality and κ -Normalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Topological properties of products of ordinals

Kemoto

Szeptycki

2004

Topology and its Applications

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“…The converse is not always true. The space u\ x ui\ + 1 is mildly normal, see [2] and [3], but not almost normal because the closed subset A = u>i x {wi} is disjoint from the regularly closed subset B = {(A, A) : a < LO\} and they cannot be separated by two disjoint open subsets, see [1].…”

Section: A Subset a Of A Topological Space X Is Called Regularly Closmentioning

confidence: 99%

On Almost Normality

Kalantan¹,

Allahabi²

2008

Demonstratio Mathematica

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Abstract.A topological space X is called almost normal if for any two disjoint closed subsets A and B of X one of which is regularly closed, there exist two open disjoint subsets U and V of X such that A C U and B C V. We will present an example of a Tychonoff almost normal space which is not normal. Almost normality is not productive. We will present some conditions to assure that the product of two spaces will be almost normal.We investigate in this paper a weaker version of normality called almost normality. We will prove that almost normality is a property which lies between mild normality and normality. We will present an example of a Tychonoff almost normal space which is not normal. We will show that almost normality is not productive and we will present some conditions to assure that the product of two spaces will be almost normal.We will denote an order pair by (x,y), the set of positive integers by N and the set of real numbers by R. A T4 space is a T\ normal space. And a Tychonoff space is a T\ completely regular space. The interior of a set A will be denoted by intyl, and the closure of a set A will be denoted by A.

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“…For instance, ω 1 × (ω 1 + 1) is mildly normal but not normal. Moreover, using elementary submodels, it is proved in [4] that products of arbitrary many ordinals are mildly normal. In [6], it is proved that for A, B ⊆ ω 1 , A × B is normal if and only if A or B is non-stationary or A ∩ B is stationary in ω 1 .…”

Section: Introductionmentioning

confidence: 99%

Mild normality of finite products of subspaces of ω1

Hirata¹,

Kemoto²

2006

Topology and its Applications

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It is known that products of arbitrary many ordinals are mildly normal [L. Kalantan, P.J. Szeptycki, κ-normality and products of ordinals, Topology Appl. 123 (2002) 537-545] and products of two subspaces of ordinals are also mildly normal [L. Kalantan, N. Kemoto, Mild normality in products of ordinals, Houston J. Math. 29 (2003) 937-947]. It was asked if products of arbitrary many subspaces of ordinals are mildly normal. In this paper, we characterize the mild normality of products of finitely many subspaces of ω 1 . Using this characterization, we show that there exist 3 subspaces of ω 1 whose product is not mildly normal.

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κ-normality and products of ordinals

Cited by 16 publications

References 9 publications

Topological properties of products of ordinals

Topological properties of products of ordinals

On Almost Normality

Mild normality of finite products of subspaces of ω1

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