Mild normality of finite products of subspaces of \(\omega_1\)
type: Item
instance of: scholarly article
title: Mild normality of finite products of subspaces of \(\omega_1\)
zbMATH Open document ID: 1091.54004
author: Yasushi Hirata
author: Nobuyuki Kemoto
published in: Topology and its Applications
publication date: 2006-03-20T00:00:00Z
review text: A topological space \(X\) is called mildly normal if every pair of disjoint regular closed sets in \(X\) can be separated by disjoint open sets. \textit{L. Kalantan} and \textit{P. J. Szeptycki} [Topology Appl. 123, 537--545 (2002; Zbl 1016.54015)] proved that the product of arbitrary many ordinals is mildly normal. \textit{L. Kalantan} and \textit{N. Kemoto} [Houston J. Math. 29, 937--947 (2003; Zbl 1155.54320)] proved that the product of two subspaces of ordinals is mildly normal, and asked if the product of arbitrary many subspaces of ordinals is mildly normal. In this paper, the authors determine when the product of finitely many subspaces of \(\omega_1\) is mildly normal. They apply the result to answer the question and show that there exist three subspaces \(A\), \(B\) and \(C\) of \(\omega_1\) such that \(A\times B\times C\) is not mildly normal. As another application, they prove that the product of finitely many subspaces of \(\omega_1\) with the pairwise non-stationary intersection is mildly normal. It, however, remains open whether the product of countable, pairwise disjoint family of stationary sets of \(\omega_1\) is mildly normal.
reviewed by: Haruto Ohta
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 5013534
zbMATH Keywords: mildly normal
zbMATH Keywords: product
zbMATH Keywords: ordinal number
zbMATH Keywords: stationary set
MaRDI profile type: Publication
full work available at URL: j.topol.2004.06.014
OpenAlex ID: W2095110377
cites work: Strong zero-dimensionality of products of ordinals.
cites work: Separating by Gδ-sets in finite powers of ω1
cites work: Q4679053
cites work: \(\kappa\)-normality and products of ordinals
cites work: Topological properties of products of ordinals
DOI: 10.1016/J.TOPOL.2004.06.014
description: scientific article; zbMATH DE number 5013534