Monotone normality and neighborhood assignments
type: Item
instance of: scholarly article
title: Monotone normality and neighborhood assignments
zbMATH Open document ID: 1247.54034
author: Hang Zhang
author: Wei-Xue Shi
published in: Topology and its Applications
publication date: 2012-04-13T00:00:00Z
review text: This paper largely deals with the proof of two Theorems, i.e., 1.1 and 1.2 and the construction of an important example, i.e., Example 3.4. The motivation for the paper are questions posed in [\textit{C. R. Borges} and \textit{A. C. Wehrly}, Topol. Proc. 16, 7--15 (1991; Zbl 0787.54023)] and [\textit{R. Z. Buzyakova, V. V. Tkachuk} and \textit{R. G. Wilson}, Commentat. Math. Univ. Carol. 48, No. 4, 689--697 (2007; Zbl 1199.54141)]. In Theorem 1.1, the authors prove that every neighborhood assignment for a monotonically normal space has a kernel which is homeomorphic to some subspace of an ordinal. As a corollary, the authors show that every monotone neighborhood assignment for a monotonically normal space has a discrete kernel (see Corollary 1.3). In Example 3.4, the authors show that it is not true that every neighborhood assignment for an arbitrary given topological space \(X\) has a kernel which is homeomorphic to some subspace of an ordinal. Indeed there is a regular space which has a neighborhood assignment with no kernels homeomorphic to any subspace of an ordinal. But the authors mention from reference [loc. cit.] that every neighborhood assignment for an arbitrary given topological space \(X\) has a scattered kernel. This is indeed a partial answer to the question whether every monotonically normal space is dually discrete (see [loc. cit.]). The authors also prove Theorem 1.2 which states that for every neighborhood assignment \(N\) of a monotonically normal space \(X\) with a monotone normality operator \(H\), there is a non-empty subspace \(D\) of \(X\) such that \(D\) is homeomorphic to some subspace of an ordinal and \(H(x, H(x, N(x))) \cap D = \emptyset\) for every \(x \in X \backslash N[D]\).
reviewed by: Saeid Jafari
Mathematics Subject Classification ID: 54D20
zbMATH DE Number: 6023675
zbMATH Keywords: monotonically normal
zbMATH Keywords: monotonically normal paracompact space
zbMATH Keywords: neighborhood assignment
zbMATH Keywords: dually discrete
zbMATH Keywords: scattered kernel
zbMATH Keywords: \(D\)-space
MaRDI profile type: Publication
full work available at URL: j.topol.2011.10.007
OpenAlex ID: W2065511773
cites work: Dually discrete spaces
cites work: Q3593901
cites work: Monotone normality
cites work: Q4276145
cites work: Q3392802
cites work: Some properties of the Sorgenfrey line and related spaces
cites work: D-spaces
cites work: Q3001417
cites work: On spaces which are linearly \(D\)
cites work: Monotonically Normal Spaces
cites work: Set theory. An introduction to independence proofs
cites work: Classes defined by stars and neighbourhood assignments
cites work: On linear neighborhood assignments and dually discrete spaces
cites work: Partition Problems in Topology
DOI: 10.1016/J.TOPOL.2011.10.007
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description: scientific article; zbMATH DE number 6023675