The Banaschewski compactification revisited (Q2318350)

The main results of the article concern constructions of the Banaschewski compactification of a zero-dimensional Hausdorff approach space. The category \(\mathbf{App}\) of approach spaces as objects and contractions as morphisms is applied. Basic facts and terminology concerning this category can be found in [\textit{R. Lowen}, Approach spaces: the missing link in the topology-uniformity-metric triad. Oxford: Clarendon Press (1997; Zbl 0891.54001)] and [\textit{R. Lowen}, Index analysis. Approach theory at work. London: Springer (2015; Zbl 1311.54002)]. The intended set-theoretic framework seems to be the same, as in [\textit{S. Mac Lane}, Categories for the working mathematician. New York-Heidelberg-Berlin: Springer (1971; Zbl 0232.18001)]. In terms of a gauge and a tower of an approach space, the authors introduce the category \(\mathbf{ZDApp}\) of zero-dimensional approach spaces as objects and contractions as morphisms. Special attention is paid to its subcategories \(\mathbf{ZDApp}_2\) of Hausdorff zero-dimensional approach spaces as objects and \(\mathbf{kZDApp}_2\) of compact Hausdorff zero-dimensional approach spaces as objects. It is noticed that \(\mathbf{ZDApp}\) is a topological category which is a fully reflective subcategory of \(\mathbf{App}\). The category \(\mathbf{ZDim}\) of zero-dimensional topological spaces with continuous maps as morphisms can be coreflectively and reflectively fully embedded in \(\mathbf{ZDApp}\). The category \(\mathbf{UMet}\) of extended pseudo-ultrametric spaces with non-expansive maps is initially dense in \(\mathbf{ZDApp}\). If \(d\) is an extended quasi-metric on \(X\), then \(\delta_d\) is the distance function between points and subsets of \(X\) defined by: \(\delta_d(x, A)=d(x, A)\) for \(x\in X\) and \(A\subseteq X\). The approach gauge \(d\downarrow\) induced by \(d\) is the collection of all extended quasi-metrics \(q\) on \(X\) such that \(\delta_q\leq \delta_d\). A crucial role is played by the extended ultrametric \(d_M\) on the interval \([0, +\infty]\) defined as follows: \(d_M(x,x)=0\) and \(d_M(x, y)=x\vee y\) for each pair of distinct \(x,y\in [0, +\infty]\). If \(X\) is an approach space, then \(\mathcal{M}_{fin}(X)\) stands for the collection of all contractions \(f:X\to([0, +\infty], d_M\downarrow)\) such that \(f(X)\) is finite. It is shown that \(\mathbf{kZDApp}_2\) is the epireflective hull in \(\mathbf{ZDApp}_2\) of the class of all finite subspaces of \(([0, +\infty], d_M\downarrow)\). Given objects \(X\) of \(\mathbf{ZDApp}_2\) and \(K\) of \(\mathbf{kZDApp}_2\), a morphism \(k_X: X\to K\) is said to be a Hausdorff zero-dimensional compactification of \(X\) if \(k_X\) is a homeomorphic embedding of \(X\) onto a dense subspace of \(K\). The authors prove that, for objects \(X\) of \(\mathbf{ZDApp}_2\) and \(K\) of \(\mathbf{kZDApp}_2\), a Hausdorff zero-dimensional compactification \(k_X:X\to K\) has the universal property with respect to all objects of \(\mathbf{kZDApp}_2\) if and only if every contraction in \(\mathcal{M}_{fin}(X)\) can be extended to a contraction on \(K\). Using this fact, by a natural embedding of an object \(X\) of \(\mathbf{ZDApp}_2\) into a product of finite subspaces of \(([0, +\infty], d_M\downarrow)\), the authors find an epireflector \(\varsigma^{\star}:\mathbf{ZDApp}_2\to\mathbf{kZDapp}_2\). For an object \(X\) of \(\mathbf{ZDApp}_2\), \(\varsigma^{\star}(X)\) is called the Banaschewski compactification of \(X\). Several other constructions of \(\varsigma^{\star}(X)\) are shown: by using appropriate uniformities and completions, by applying proximities and relevant Smirnov's compactifications, by applying the Wallman-Frink method to normal bases, as the component space of the Čech-Stone compactification \(\beta^{\star}(X)\) of \(X\). The constructions are adapted to the case when \(X\) is a topological zero-dimensional Hausdorff space or an (extended) ultrametric space. In particular, given an (extended) ultrametric space \((X, d)\) and the related distance \(\delta_d\) between points and subsets of \(X\), an extension \((Z, \delta^{\varsigma})\) is constructed where \(Z\) is the underlying set of the Smirnov compactification of the proximity on \(X\) induced by \(d\), and \(d^{\varsigma}\) is a distance between points and subsets of \(Z\) such that \(\delta^{\varsigma}\) is an extension of \(\delta_d\), and \(Z\) equipped with the topology induced by \(\delta^{\varsigma}\) is a Hausdorff zero-dimensional compactification of \(X\) equipped with the topology induced by \(d\). The authors remark that \((Z, \delta^{\varsigma})\) retains numerical information from \((X, d)\), while the topological Banaschewski compactification of \(X\) equipped with the topology induced by \(d\) need not be metrizable. Finally, it is proved that \(\varsigma^{\star}(X)\) and \(\beta^{\star}(X)\) coincide for every ultrametric space \(X\).