A little more on coz-unique frames
type: Item
instance of: scholarly article
title: A little more on coz-unique frames
zbMATH Open document ID: 1175.06005
author: Themba Dube
published in: Applied Categorical Structures
publication date: 2009-09-25T00:00:00Z
review text: A frame is \textit{coz-unique} in case its cozero part is the only regular sub-\(\sigma\)-frame generating it. Coz-unique frames were introduced and characterized by \textit{B. Banaschewski} and \textit{C. Gilmour} in [``Cozero bases of frames'', J. Pure Appl. Algebra 157, No. 1, 1--22 (2001; Zbl 0964.54020)]. In the paper under review, the author gives further characterizations of these frames. These results are extensions to frames of the following characterizations of absolutely \(z\)-embedded spaces obtained, respectively, by \textit{R. L. Blair} and \textit{A. W. Hager} [``Extensions of zero-sets and of real-valued functions'', Math. Z. 136, 41--52 (1974; Zbl 0264.54011)] and by \textit{A. W. Hager} and \textit{D. G. Johnson} [``A note on certain subalgebras of \(C(\mathfrak{X})\)'', Can. J. Math. 20, 389--393 (1968; Zbl 0162.26702)]: {\parindent=4mm \begin{itemize}\item[--] For every completely regular space \(X\) the following are equivalent: (1) \(X\) is absolutely \(z\)-embedded; (2) \(X\) is \(z\)-embedded wherever it is embedded; (3) \(X\) is either Lindelöf or almost compact; (4) If \(Z_1\) and \(Z_2\) are disjoint zero-sets of \(X\), either \(Z_1\) is compact or \(Z_2\) is Lindelöf; (5) If \(Z\) is a zero-set of \(X\), either \(Z\) is compact or \(X\setminus Z\) is Lindelöf. \item[--] Every embedding of a Tychonoff space \(X\) is a \(z\)-embedding if and only if the only realcompact space in which \(X\) is \(G_{\delta}\)-dense is the Hewitt realcompactification \(v X\). \end{itemize}} The spatial notion of \(z\)-embedded subspaces is captured in the point-free setting by coz-onto frame homomorphisms (that is, morphisms that map cozero parts of their domains onto cozero parts of their codomains). The paper ends with characterizations of these maps in terms of normal covers, along the lines of [\textit{R. A. Alò, L. Imler} and \textit{H. L. Shapiro}, Math. Ann. 188, 13--22 (1970; Zbl 0188.55202)].
reviewed by: Jorge Picado
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 54C25
Mathematics Subject Classification ID: 54D20
zbMATH DE Number: 5607534
zbMATH Keywords: cozero element
zbMATH Keywords: cozero part
zbMATH Keywords: coz-onto frame homomorphism
zbMATH Keywords: coz-codense frame homomorphism
zbMATH Keywords: coz-unique frame
zbMATH Keywords: \(C\)-quotient
zbMATH Keywords: \(C^*\)-quotient
MaRDI profile type: Publication
full work available at URL: s10485-008-9125-8
OpenAlex ID: W1977265682
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DOI: 10.1007/S10485-008-9125-8
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description: scientific article; zbMATH DE number 5607534