Direct limit topologies of quasi-uniform spaces and paratopological groups
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 \((G,\tau)\) is a paratopological group and \(\tau^*\) is the coarsest group topology on \(G\) that is finer than \(\tau\), then \(\tau^*\) is identical with the finest topology on \(G\) that makes all inclusion maps \((G_n,\tau^*_n)\hookrightarrow G\) continuous. They also consider four more topologies on \(G\) and provide conditions on \(((G_n,\tau_n))_{n<\omega}\) under which they turn \(G\) into a paratopological group, and in particular under which they are identical with \(\tau\). For topological groups, these topologies have been studied by \textit{T. Banakh} and \textit{D. Repovš} [Tohoku Math. J. (2) 64, No. 1, 1--24 (2012; Zbl 1241.22002)].
scientific article; zbMATH DE number 6495772