Nonshrinking open covers and K. Morita's third conjecture
type: Item
instance of: scholarly article
title: Nonshrinking open covers and K. Morita's third conjecture
zbMATH Open document ID: 0938.54025
DOI: 10.1016/S0166-8641(97)00091-6
published in: Topology and its Applications
publication date: 2000-06-18T00:00:00Z
review text: \textit{K. Morita} conjectured [Gen. Topol. Relat. Mod. Anal. Algebra, Proc. 4th Prague Topol. Symp. 1976, Part B, 296-297 (1977; Zbl 0374.54004)] that a space \(X\) is a \(\sigma\)-locally compact, metrizable space if and only if \(X \times Y\) is normal for every normal, countably paracompact space \(Y\). Here a space is said to be \(\sigma\)-locally compact if it is the union of countably many closed locally compact subspaces. In this paper, his conjecture is proved by showing that for every regular uncountable cardinal \(\kappa\), there is a hereditarily normal, countably paracompact space \(X_{\kappa}\) with an increasing open cover \({\mathcal W} = \{ W_{\mu}: \mu \in \omega_1 \}\) such that \(\mathcal W\) does not have any refinement by \(\leq \kappa\) many closed sets. It was proved by \textit{K. Chiba, T. C. Przymusiński} and \textit{M. E. Rudin} [Topology Appl. 22, 19-32 (1986; Zbl 0575.54007)] that the existence of such spaces implies Morita's conjecture. The spaces \(X_{\kappa}\) are the first known examples of hereditarily normal \(\omega_1\)-Dowker spaces. The following conjecture (Morita's second conjecture) remains open in ZFC: A space is metrizable if and only if the product of the space with every normal Morita \(P\)-space is normal.
reviewed by: Ken-ichi Tamano
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 54F65
zbMATH DE Number: 1308031
zbMATH Keywords: \(\sigma\)-locally compact metrizable space
zbMATH Keywords: product
zbMATH Keywords: normal
zbMATH Keywords: countably paracompact
zbMATH Keywords: shrinking
zbMATH Keywords: \(\omega_1\)-Dowker space
Wikidata QID: Q122997965
author: Z. Balogh
MaRDI profile type: Publication
cites work: Q3477647
cites work: A small Dowker space in ZFC
cites work: Set-theoretic constructions of nonshrinking open covers
cites work: Normality of product spaces and Morita's conjectures
cites work: On Countably Paracompact Spaces
cites work: Set theory. An introduction to independence proofs
cites work: Q3337413
cites work: Products of normal spaces with metric spaces
cites work: Q5185672
cites work: $\aleph $-Dowker spaces
cites work: Q3681543
description: scientific article; zbMATH DE number 1308031