Monotone countable paracompactness
type: Item
instance of: scholarly article
title: Monotone countable paracompactness
zbMATH Open document ID: 0938.54026
DOI: 10.1016/S0166-8641(98)00128-X
author: Robin W. Knight
author: I. S. Stares
author: Chris Good
published in: Topology and its Applications
publication date: 2000-06-18T00:00:00Z
review text: The authors define and study a monotone version of countable paracompactness. They opt to monotonize Ishikawa's characterization and arrive at: a space is Monotonically Countably Paracompact if one can assign to each decreasing sequence \(F=\langle F_n\rangle_n\) of closed sets with empty intersection a sequence \(U_F=\langle U_{F,n}\rangle_n\) of open sets, with \(\bigcap_n \operatorname{cl}U_{F,n}=\emptyset\), such that \(F_n\subseteq U_{F,n}\) for all \(n\) (this characterizes countable paracompactness) and if \(F_n\subseteq G_n\) for all \(n\) then \(U_{F,n}\subseteq U_{G,n}\) for all \(n\) (this is the monotonicity requirement) -- demanding that only \(\bigcap_nU_{F,n}=\emptyset\) leads to Monotone Countable Metacompactness. MCP and MCM spaces turn out to be closely connected to the \(\beta\)- and wN-spaces of \textit{R. E. Hodel} [Duke Math. J. 39, 253-263 (1972; Zbl 0242.54027)]; in fact MCM spaces are the same as \(\beta\)-spaces and every wN-space is MCP (the converse is true under mild assumptions, like local countable compactness). A consequence is the quite unexpected result that the familiar Sorgenfrey and Michael lines are not MCM; also the lexicographically ordered group \(\mathbb{Z}^{\omega_1}\) is not MCP. Because \(X\) is stratifiable iff the product \(X\times[0,1]\) is monotonically normal there is no monotone parallel of Dowker's characterization of normal countably paracompact spaces.
reviewed by: K. P. Hart
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54E20
Mathematics Subject Classification ID: 54E30
zbMATH DE Number: 1395043
zbMATH Keywords: monotone normality
zbMATH Keywords: \(\beta\)-space
zbMATH Keywords: wN-space
zbMATH Keywords: stratifiability
MaRDI profile type: Publication
full work available at URL: s0166-8641(98)00128-x
OpenAlex ID: W2056426457
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description: scientific article; zbMATH DE number 1395043