A complete classification of the simultaneously bireflective and bicoreflective subconstructs of approach spaces
Approach spaces have been studied as a common generalization of topological and metric spaces respectively [\textit{R. Lowen}, Approach spaces: the missing link in the topology-uniformity-metric triad (1997; Zbl 0891.54001)]. They form a topological category \textbf{AP} in the sense of Herrlich which contains the category \textbf{Top} of topological spaces as a simultaneously bireflective and bicoreflective (concrete) subcategory. In view of the fact that most familiar topological categories do not admit any non-trivial simultaneously bireflective and bicoreflective (concrete) subcategory [\textit{V. Kannan}, Math. Ann. 195, 168-174 (1972; Zbl 0215.09701)] for the case of \textbf{Top}) the last mentioned property of \textbf{AP} can be considered surprising and leads the authors to determine all such subcategories of \textbf{AP}. These are shown to form a family \({\mathbf A}{\mathbf P}_m\), \(m\in \overline\mathbb{R}\), \(0\leq m\leq \infty\) of categories with \({\mathbf A}{\mathbf P}_0={\mathbf A}{\mathbf P}\), \({\mathbf A}{\mathbf P}_\infty=\) \textbf{Top} and \(|{\mathbf A}{\mathbf P}_p|\subset |{\mathbf A}{\mathbf P}_q|\) for \(p<q\). For each \(m\) the objects of \({\mathbf A}{\mathbf P}_m\) are characterized in terms of approach structures which finally leads to the observation that, for \(0<p\), \(q<\infty\), the categories \({\mathbf A}{\mathbf P}_p\) and \({\mathbf A}{\mathbf P}_q\) are isomorphic.
scientific article; zbMATH DE number 1563344