Absorbing spaces for \(C\)-compacta
type: Item
instance of: scholarly article
title: Absorbing spaces for \(C\)-compacta
zbMATH Open document ID: 0930.54027
DOI: 10.1016/S0166-8641(97)00103-X
published in: Topology and its Applications
publication date: 2000-02-13T00:00:00Z
review text: Recall that a space \(X\) is a \(C\)-space if each sequence \(\{\alpha_n\mid n\in\mathbb N\}\) of open covers admits an open cover which can be written as the countable union of pairwise disjoint collections \(\beta_n\) where each \(\beta_n\) refines \(\alpha_n\) [\textit{D. Addis} and \textit{J. Gresham}, Fundam. Math. 101, 195-205 (1978; Zbl 0397.54051)]. There is a notion \(\dim_C\), which is the transfinite extension of covering dimension that classifies \(C\)-compacta (spaces here are separable and metrizable). This concept was first defined by P. Borst and is recaptured in the current paper. The author proves that for every countable ordinal \(\alpha\), there exists a \(C\)-compactum which is universal for the class of all compacta \(X\) with \(\dim_C X\leq\alpha\). The result is applied to show that for uncountably many ordinals \(\beta\), there exist noncountable-dimensional pre-Hilbert spaces \(D_\beta\) which are absorbing spaces [\textit{M. Bestvina} and \textit{J. Mogilski}, Mich. Math. J. 33, 291-313 (1986; Zbl 0629.54011)] for the class of compacta with \(\dim_C<\beta\).
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 57N17
Mathematics Subject Classification ID: 57N20
zbMATH DE Number: 1308044
zbMATH Keywords: absorbing space
zbMATH Keywords: pre-Hilbert space
zbMATH Keywords: Brouwer-Kleene order
zbMATH Keywords: analytic set
zbMATH Keywords: weakly infinite-dimensional
zbMATH Keywords: countable-dimensional
zbMATH Keywords: \(C\)-space
author: Taras Radul
reviewed by: Leonard R. Rubin
MaRDI profile type: Publication
cites work: Some remarks concerning C-spaces
cites work: Characterizing certain incomplete infinite-dimensional absolute retracts
cites work: Sigma-Compact Locally Convex Metric Linear Spaces Universal for Compacta are Homeomorphic
cites work: Q4014763
cites work: A class of infinite-dimensional spaces. Part I: Dimension theory and Alexandrof's problem
cites work: Q3469885
cites work: Q3363960
cites work: On classification of weakly infinite-dimensional compacta
cites work: Q5344739
cites work: Q4060144
cites work: Extending Homeomorphisms and Applications to Metric Linear Spaces Without Completeness
cites work: A Weakly Infinite-Dimensional Compactum which is not Countable-Dimensional
cites work: Q3961484
full work available at URL: s0166-8641(97)00103-x
OpenAlex ID: W2087836919
description: scientific article; zbMATH DE number 1308044