Approach theory in a category: A study of compactness and Hausdorff separation
A topological construct \({\mathcal X}\) with a proper \(({\mathcal E},{\mathcal M})\)-factorization system is endowed with a distance operator by defining a concrete functor \(\Lambda:{\mathcal X}\rightarrow { Prap}\) to the construct \(Prap\) of pre-approach spaces and contractions. Then the following class of ``closed morphisms'' arises naturally: \({\mathcal F}_{\Lambda}=\{f: \Lambda f\) is a closed contraction in \(Prap\}.\) For \(\Lambda\) preserving subobjects, all the axioms put forward in the work of \textit{M. M. Clementino, E.Giuli} and \textit{W. Tholen} [Pedicchio, Maria Cristina (ed.) et al., Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Cambridge: Cambridge University Press. Encycl. Math. Appl. 97, 103--163 (2004; Zbl 1059.54012)] in order to develop a functional approach to general topology with respect to the class \({\mathcal F}_{\Lambda}\) are fulfilled. The author concentrates on \({\mathcal F}_{\Lambda}\)-compactness of an object \(\underline{X}\) in \({\mathcal X},\) which expresses the fact that the projection \(p_Y:{\underline{X}}\times {\underline{Y}}\rightarrow \underline{Y}\) belongs to \({\mathcal F}_{\Lambda}\) for every \({\mathcal X}\)-object \(\underline{Y}.\) Under certain conditions on \(\Lambda\), she proves that \({\mathcal F}_\Lambda\)-compactness of \(\underline{X}\) is equivalent to \(0\)-compactness of \(\Lambda(\underline{X})\) in \(Prap\). She finds that for \(\Lambda\) preserving subobjects, an \({\mathcal X}\)-object \(\underline{X}\) is \({\mathcal F}_{\Lambda}\)-Hausdorff iff the pretopological reflection of \(\Lambda(\underline{X})\) is Hausdorff. In \(Top\) the classical compactness and \(b\)-compactness are captured if for \(\Lambda\) the functor describing the Kuratowski or \(b\)-closure, respectively, is considered. On the construct \(Ap\) of approach spaces natural functors \(\Lambda\) arise, which cannot be described by means of a closure operator.
scientific article; zbMATH DE number 5324371