Direct limit topologies of quasi-uniform spaces and paratopological groups
type: Item
instance of: scholarly article
title: Direct limit topologies of quasi-uniform spaces and paratopological groups
zbMATH Open document ID: 1335.22003
DOI: 10.1215/17358787-3163248
author: Fucai Lin
author: Li-Hong Xie
author: Jing Zhang
published in: Banach Journal of Mathematical Analysis
publication date: 2015-10-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bjma/1444913862
review text: Let \((X_n)_{n<\omega}\) be a sequence of quasi-uniform spaces such that \(X_0\subset X_1\subset X_2\subset\cdots\subset X_n\subset\cdots\) and let \(X\) be the corresponding direct limit, i.e., \(X=\bigcup_{n<\omega}X_n\) and \(X\) has the strongest quasi-uniformity that makes all inclusion maps \(X_n\hookrightarrow X\) quasi-uniformly continuous. The authors show that this quasi-uniformity is generated by a family of quasi-pseudo-metrics on \(X\), which they construct as follows. For each \(n<\omega\), let \({\mathcal P}_n\) be the family of all quasi-uniform quasi-pseudo-metrics on \(X_n\). A sequence \((d_n)_{n<\omega}\in \prod_{b<\omega}{\mathcal P}_n\) is said to be monotone if \(d_n\leq d_{n+1}\mid(X_n\times X_n)\) for each \(n<\omega\). Every monotone sequence \((d_n)_{n<\omega}\in\prod_{n<\omega}{\mathcal P}_n\) induces a quasi-pseudo-metric \(\varinjlim d_n\) on \(X\) defined by \(\varinjlim d_n(x,y)= \inf\{\sum^n_{i=1} d_{|x_{i-1},x_i|}(x_{i-1}, x_i)\mid x= x_0,x_1,\dots, x_n= y\}\), where \(|x_{i-1}, x_i|= \max\{|x_{i-1}|,|x_i|\}\) and \(|x_{i-1}|= \min\{n<\omega\mid x_{i-1}\in X_n\}\) and \(|x_i|= \min\{n<\omega\mid x_i\in X_n\}\). The resulting family \(\{\varinjlim d_n\mid(d_n)_{n<\omega}{\mathcal P}_n\) is monotone\} of quasi-pseudo-metrics on \(X\) is shown to generate the quasi-uniformity of the direct limit \(X\). The authors also investigate sequences \(((G_n,\tau_n))_{n<\omega}\) of paratopological groups such that \(G_0\subset G_1\subset G_2\subset\cdots\subset G_n\subset\cdots\) and their direct limits, where a paratopological group is a group with a topology such that the product map is jointly continuous. In general, the finest topology \(\tau\) on \(G= \bigcup_{n<\omega} G_n\) that makes all inclusion maps \((G_n,\tau_n)\hookrightarrow G\) continuous does not turn \(G\) into a paratopological group. For each \(n<\omega\), let \(\tau^*_n\) be the coarsest group topology on \(G_n\) that is finer than \(\tau_n\). The authors show that, if \((G…
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 18A30
zbMATH DE Number: 6495772
zbMATH Keywords: direct limit
zbMATH Keywords: quasi-uniform space
zbMATH Keywords: quasi-pseudo-metric
zbMATH Keywords: paratopological group
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
OpenAlex ID: W2103475679
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description: scientific article; zbMATH DE number 6495772