Smyth completion as bicompletion
In his thesis \textit{H. J. K. Junnila} defined the well-monotone quasi-uniformity \({\mathcal W}(X) \) of a topological space \(X\). \textit{N. Ferrario} and \textit{H. P. A. Künzi} [Math. Proc. Camb. Philos. Soc. 109, No. 1, 167-186 (1991; Zbl 0731.54021)] then showed that the sobrification of a \(T_0\)-space \(X\) can be constructed as the topological space underlying the bicompletion of \((X,{\mathcal W}(X)).\) The authors of the present paper introduce a functor \({\mathbf w}\) from the category of topological quasi-uniform \(T_0\)-spaces and continuous, uniformly continuous maps (in the sense of Smyth-Sünderhauf) to the category of quasi-uniform \(T_0\)-spaces and uniformly continuous maps that equips the underlying set \(X\) of a topological quasi-uniform space \((X,{\mathcal U},{\mathcal T})\) with some kind of well-monotone quasi-uniformity \({\mathcal U}_{{\mathcal T}}\) derived from \({\mathcal U}\) and \({\mathcal T}\) and that acts as the identity on morphisms. They show that \(\tau({\mathcal U}_{{\mathcal T}})={\mathcal T}\) and \(\tau(({\mathcal U}_{\mathcal T})^{-1})=\tau({\mathcal U}^{-1})\) and that in a certain sense \({\mathcal U}\) can be recovered from \({\mathcal U}_{{\mathcal T}}.\) Moreover, by developing the methods of Ferrario and Künzi, they prove that for any topological quasi-uniform space \((X,{\mathcal U},{\mathcal T})\) with Smyth completion \((\widetilde{X},\widetilde{{\mathcal U}},\widetilde{{\mathcal T}})\) the roundification \(\text{Rd}\) of filters yields a uniform isomorphism from the bicompletion \((\widehat{X},\widehat{{\mathcal U}_{{\mathcal T}}})\) of \((X,{\mathcal U}_{{\mathcal T}})\) to \((\widetilde{X},\widetilde{\mathcal U}_{\widetilde{{\mathcal T}}}).\) Indeed, they verify that \(\text{Rd}\) is a natural isomorphism \({\mathbf b}\circ{\mathbf w}\rightarrow {\mathbf w}\circ {\mathbf s}\) where \({\mathbf b}\) (resp. \({\mathbf s})\) denotes the bicompletion (resp. Smyth completion) functor. In particular, it follows from their results that the Smyth completion \((\widetilde{X},\widetilde{{\mathcal U}},\widetilde{{\mathcal T}})\) of \((X,{\mathcal U},{\mathcal T})\) can be obtained via the bicompletion \((\widehat{X},\widehat{{\mathcal U}_{{\mathcal T}}})\) and that a topological quasi-uniform space is Smyth complete exactly when its image under the functor \({\mathbf w}\) is bicomplete. \{For a filter \({\mathcal F}\) on a topological quasi-uniform space \((X,{\mathcal U},{\mathcal T})\), \(\text{Rd}({\mathcal F})= \{B\subseteq X:\) There are \(A\in {\mathcal F}\) and \(U\in {\mathcal U}\) such that \(A \prec_U B\}\) is its roundification. The casual reader is warned that the authors use \(U(x)\) to denote sets that are usually called \(U^{-1}(x)\) in similar articles.\}.
scientific article; zbMATH DE number 1307898
scientific article