\(\kappa\)-normality and products of ordinals
type: Item
instance of: scholarly article
title: \(\kappa\)-normality and products of ordinals
zbMATH Open document ID: 1016.54015
DOI: 10.1016/S0166-8641(01)00220-6
author: Lutfi Kalantan
author: Paul J. Szeptycki
published in: Topology and its Applications
publication date: 2003-03-10T00:00:00Z
review text: Let \(Y\) be a subspace of a topological space \(X\). Following \textit{A. V. Arhangel'skii} [Topology Appl. 70, 87-99 (1996; Zbl 0848.54016)] a subset \(A\) of \(X\) is said to be concentrated on \(Y\) if \(A\subset \text{cl}_X(A\cap Y)\). \(X\) is said to be normal on \(Y\) if every two disjoint closed subsets of \(X\) concentrated on \(Y\) can be separated by disjoint open neighborhoods in \(X\). \(X\) is called densely normal if there exists a dense subspace \(Y\) of \(X\) such that \(X\) is normal on \(Y\). Every densely normal space is \(\kappa\)-normal in the sense of \textit{E. V. Shchepin} [Sib. Math. J. 13, No. 5, 820-830 (1972; Zbl 0261.54014)]. \textit{W. Just} and \textit{J. Tartir} have shown that there exist Tychonoff spaces which are densely normal but not \(\kappa\)-normal [Proc. Am. Math. Soc. 127, No. 3, 901-905 (1999; Zbl 0907.54011)]. In fact, as \textit{A. V. Arhangel'skii} has recently pointed out, the space \(X= C_p(\omega_1+ 1)\) is \(\kappa\)-normal, but no dense subspace \(Y\) of \(X\) is densely normal [Topology Appl. 123, No. 1, 27-36 (2002; Zbl 1008.54013)]. Here products of ordinals are considered and the following results are obtained. Theorem 1: Suppose that \(\alpha_i\) is an ordinal for each \(i<\omega\). Then \(\prod\{\alpha_i\mid i< \omega\}\) is densely normal. Theorem 2: Suppose that \(\alpha_i\) is an ordinal for each \(i<\lambda\). Then \(\prod\{\alpha_i\mid i<\lambda\}\) is \(\kappa\)-normal. The proof of Theorem 2 uses extensively the technique of elementary submodels.
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 1879073
zbMATH Keywords: densely normal space
zbMATH Keywords: \(\kappa\)-normal space
zbMATH Keywords: regular open
zbMATH Keywords: ordinals
zbMATH Keywords: normality
zbMATH Keywords: countable paracompactness
zbMATH Keywords: elementary submodels
zbMATH Keywords: products
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
cites work: Relative topological properties and relative topological spaces
cites work: Q4150804
cites work: A 𝜅-normal, not densely normal Tychonoff space
cites work: Q4372664
cites work: Results about \(\kappa\)-normality
cites work: Countably compact extensions of topological spaces
cites work: Correction to ''Countably compact extensions of topological spaces''
cites work: Q5185672
cites work: Real functions and near-normal spaces
cites work: ON $ \kappa$-METRIZABLE SPACE
cites work: Paracompactness and product spaces
cites work: Q5653336
Recommended article: \(\beta\)-normality in products of two ordinals
Recommended article: Q4679053
Recommended article: Normal subspaces of products of finitely many ordinals
Recommended article: The normality of \(\Sigma\)-products and the perfect \(\kappa\)-normality of Cartesian products
Recommended article: Q4369851
Recommended article: Q4204776
Recommended article: Q4907396
Recommended article: Topological properties of products of ordinals
Recommended article: Q4344960
Recommended article: Q4316899
description: scientific article; zbMATH DE number 1879073