Uniform universal covers of uniform spaces
type: Item
instance of: scholarly article
title: Uniform universal covers of uniform spaces
zbMATH Open document ID: 1116.54016
author: Valera Berestovskii
author: Conrad Plaut
published in: Topology and its Applications
publication date: 2007-05-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0607353
review text: The authors develop a generalized covering space theory for a large class of uniform spaces, called coverable spaces, including geodesic metric spaces and connected and locally pathwise connected compact topological spaces, in particular Peano continua and the topological sine curve. The idea is the following : Let \(E\) be an entourage of \(X\). An \(E\)-chain between \(p\) and \(q\) is an ordered set of points \(x_0= p, x_1, \ldots, x_n =q\) such that for all \(i\), \((x_i, x_{i+1})\in E\). Two \(E\)-chains are equivalent if the first one can be obtained from the second one through finitely many steps consisting of adding or removing a point. We denote by \(X_E\) the equivalence classes of \(E\)-chains starting at a fixed point and \(\delta_E\) the group of classes starting and leaving at the fixed point. If \(F\subset E\), we have a natural map \(X_F\to X_E\), and we denote by \(\tilde X = \displaystyle\lim_{\leftarrow} X_E\) and \(\delta_1 = \displaystyle\lim_{\leftarrow} \delta_E\). The uniform covering space \(\tilde X\) of a coverable space (to be defined) is a kind of generalized cover with deck group \(\delta_1\), and with universal and lifting properties that generalize the classical topological situation.
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 57M10
zbMATH DE Number: 5151663
zbMATH Keywords: uniform spaces
zbMATH Keywords: covering space
zbMATH Keywords: Peano continua
zbMATH Keywords: coverable spaces
reviewed by: Yves Félix
MaRDI profile type: Publication
OpenAlex ID: W1965315459
arXiv ID: math/0607353
cites work: Covering group theory for topological groups
cites work: Covering group theory for locally compact groups
cites work: Retracted: The topological fundamental group and generalized covering spaces
cites work: Q5522002
cites work: The combinatorial structure of the Hawaiian earring group
cites work: The big fundamental group, big Hawaiian earrings, and the big free groups
cites work: On the fundamental groups of one-dimensional spaces
cites work: Some Point-Set Properties and the Edge Path Group of a Generalized Uniform Space
cites work: The fundamental groups of one-dimensional spaces and spatial homomorphisms
cites work: The Fundamental Group of One-Dimensional Spaces
cites work: Infinite Products of Semi-Groups and Local Connectivity
cites work: Q5327708
cites work: Q3027866
cites work: Q5541833
cites work: Theory of Covering Spaces
cites work: Noncollapsing examples with positive Ricci curvature and infinite topological type
cites work: A Van Kampen Theorem for Weak Joins
cites work: Q3293867
cites work: Quotients of uniform spaces
cites work: Metric spaces of curvature \(\geq k\)
cites work: THE FUNDAMENTAL GROUP OF THE HAWAIIAN EARRING IS NOT FREE
cites work: Hausdorff convergence and universal covers
cites work: Universal covers for Hausdorff limits of noncompact spaces
cites work: Q5522742
DOI: 10.1016/J.TOPOL.2006.12.012
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description: scientific article