Hausdorff separation in categories
The basis for the investigations in this paper consists of a category \({\mathcal X}\), a class \({\mathcal M}\) of morphisms in \({\mathcal X}\) closed under composition and containing all isomorphisms, and \({\mathcal X}\) is assumed to be \({\mathcal M}\)-complete. This implies that there is a unique class of morphisms \({\mathcal E}\subset {\mathcal X}\), such that \(({\mathcal E},{\mathcal M})\) is a factorization system. As an additional parameter a class \({\mathcal P}\) of \({\mathcal X}\)-objects closed under morphisms of \({\mathcal E}\) is introduced. For any object \(X\) the class of points \(pt(X)\) consists of those subobjects of \(X\) with domain in \({\mathcal P}\). As a third parameter a closure operator \(C\) with respect to \({\mathcal M}\), as defined by \textit{D. Dikranjan} and \textit{E. Giuli} [Topology Appl. 27, 129-143 (1987; Zbl 0634.54008)], is considered in \({\mathcal X}\). With respect to these three parameters \({\mathcal M}\), \({\mathcal P}\) and \(C\), \(T_ 2\)-objects in \({\mathcal X}\) are defined and the full subcategory of \(T_ 2\)-objects is studied. Under a smallness condition imposed on \({\mathcal P}\), more precisely on \(pt(X)\) for any object \(X\), one gets a canonical functor \(Pt\) from the category of \(T_ 2\)-objects to the category of closure spaces. Under natural conditions this functor preserves and reflects \(T_ 2\)-objects, which implies that the category of \(T_ 2\)-objects of \({\mathcal X}\) is closed under monosources. In the last section, connections between the subcategory of \(T_ 2\)-objects and the subcategory of separated objects of \({\mathcal X}\), defined by \(C\)-closed diagonals are investigated. The introduction of the additional parameter \({\mathcal P}\) allows a wider range of applications and a finer analysis of the notion \(T_ 2\) than former investigations, which used the absolute notion of ``constant''.
constant morphism with respect to a class
scientific article; zbMATH DE number 554354