An application of Mary Ellen Rudin's solution to Nikiel's conjecture
type: Item
instance of: scholarly article
title: An application of Mary Ellen Rudin's solution to Nikiel's conjecture
zbMATH Open document ID: 1333.54032
author: Paul Cairns
author: Heikki J. K. Junnila
author: Peter Nyikos
published in: Topology and its Applications
publication date: 2015-11-09T00:00:00Z
review text: Following H. Junnila, a regular \(T_1\)-space \(X\) is called utterly normal if it has a magnetic base system, i.e., a collection \(\{{\mathcal B}_x\mid x\in X\}\), where each \({\mathcal B}_x\) is a base for the neighborhoods of \(x\), such that if \(B_x\in{\mathcal B}_x\) and \(B_y\in{\mathcal B}_y\) and \(B_x\cap B_y\neq\emptyset\), then either \(x\in\text{cl\,}B_y\) or \(y\in\text{cl\,}B_x\). Every utterly normal space is monotonically normal in the sense of \textit{R. W. Heath} et al. [Trans. Am. Math. Soc. 178, 481--493 (1973; Zbl 0269.54009)]. Whether the converse is true, i.e., whether every monotonically normal space is utterly normal, is unknown. It is also an open problem whether every utterly normal space has an open magnetic base system, i.e., a magnetic base system consisting of open sets. In light of these questions the main results of this paper are particularly interesting. It is first shown that every suborderable space is an utterly normal space with an open magnetic base system. Since, by M. E. Rudin's solution to Nikiel's conjecture, every compact monotonically normal space is the continuous image of a compact linearly ordered space, it follows that every compact monotonically normal space is utterly normal with an open magnetic base system. This fact is then used to prove that every compact monotonically normal space is hereditarily weakly orthocompact in the sense of \textit{B. M. Scott} [Stud. Topol., Proc. Conf. Charlotte, N. C., 1974, 517--537 (1975; Zbl 0311.54023)]. It follows that the one point compactification of a locally compact monotonically normal space \(X\) is monotonically normal if and only if \(X\) is weakly orthocompact. Hence every locally compact, weakly orthocompact, monotonically normal space is utterly normal with an open magnetic base system. It is unknown whether every stratifiable space is utterly normal, but the authors prove that every stratifiable space with an open magnetic base system is an \(M_1\) space.
reviewed by: Harald Brandenburg
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D45
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 54E99
zbMATH DE Number: 6506268
zbMATH Keywords: utterly normal space
zbMATH Keywords: monotonically normal space
zbMATH Keywords: suborderable space
zbMATH Keywords: weakly orthocompact space
zbMATH Keywords: \(M_1\) space
MaRDI profile type: Publication
full work available at URL: j.topol.2015.09.023
OpenAlex ID: W1832938924
Wikidata QID: Q123189548
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DOI: 10.1016/J.TOPOL.2015.09.023
description: scientific article; zbMATH DE number 6506268