On the spectral height of \(F\)-compact spaces
type: Item
instance of: scholarly article
title: On the spectral height of \(F\)-compact spaces
zbMATH Open document ID: 1325.54006
DOI: 10.1134/S0037446613030026
author: M. A. Baranova
author: A. V. Ivanov
published in: Siberian Mathematical Journal
publication date: 2013-08-12T00:00:00Z
review text: The motivation for the study of \(F\)-compact spaces, introduced in [\textit{A. V. Ivanov}, in: Mappings and functors, Work Collect., Moskva 1984, 31--40 (1984; Zbl 0565.54015)], comes from Fedorchuk's method of fully closed mappings -- see e.g. [\textit{V. V. Fedorchuk}, Russ. Math. Surv. 35, No. 3, 131--143 (1980; Zbl 0529.54008); translation from Usp. Mat. Nauk 35, No. 3, 112--121 (Russian) (1980)] and [\textit{V. V. Fedorchuk}, Fundam. Prikl. Mat. 9, No. 4, 105--235 (2003); translation in J. Math. Sci., New York 136, No. 5, 4201--4292 (2006; Zbl 1073.54010)]. A fully closed mapping is a continuous mapping \(f:X\rightarrow Y\) such that, for every \(y\in Y\) and every finite cover \(\{U_1,\dots,U_n\}\) of \(f^{-1}[\{y\}]\) by open subsets of \(X\), the set \(\{y\}\cup\bigcup_{i=1}^{n}f^{\#}U_i\) is open in \(Y\); here, \(f^{\#}A=\{y\in Y:f^{-1}[\{y\}]\subseteq A\}\) is the small image of \(A\) under \(f\). A compact Hausdorff space is said to be an \(F\)-compact space if it is the limit of a well-ordered continuous inverse system \(\{X_{\alpha},p^{\alpha'}_{\alpha}:\alpha,\alpha'<\beta\}\) of compact Hausdorff spaces such that \(|X_0|=1\) and, for each \(\alpha\) with \(\alpha+1<\beta\), the bonding mapping \(p^{\alpha+1}_{\alpha}\) is fully closed and all of its preimages \((p^{\alpha+1}_{\alpha})^{-1}[\{x\}]\) for \(x\in X_\alpha\) are metrizable. The spectral height of an \(F\)-compact space \(X\), denoted by \(sh(X)\), is defined as the least length \(\beta\) of an inverse system that satisfies the above conditions and has limit \(X\). It follows from this definition that, if \(\gamma\) is a limit ordinal, then no \(F\)-compact space \(X\) satisfies \(sh(X)=\gamma+1\). In this paper, the authors show that, for every ordinal \(\beta\in(0,\omega_1]\) that is not the successor of a limit ordinal, there is an \(F\)-compact space \(X\) with \(sh(X)=\beta\). The construction makes use of the method of resolutions, also due to Fedorchuk see e.g. [loc. cit.].
reviewed by: Rodrigo R. Dias
Mathematics Subject Classification ID: 54B35
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54B15
Mathematics Subject Classification ID: 54B99
zbMATH DE Number: 6197547
zbMATH Keywords: fully closed mapping
zbMATH Keywords: \(F\)-compact space
zbMATH Keywords: spectral height
zbMATH Keywords: resolution
MaRDI profile type: Publication
cites work: Q4281196
cites work: Q3545609
cites work: Degree of discrete generation of compact sets
cites work: A generalization of Gruenhage's example
cites work: Fully closed maps, scannable spectra and cardinality of hereditarily separable spaces
cites work: Hereditary normality of F-bicompacta
cites work: Q5580156
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full work available at URL: s0037446613030026
OpenAlex ID: W1972045563
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description: scientific article; zbMATH DE number 6197547