In an abstract category with suitable notions of subobject, closure and point, we discuss the separation axioms and . Each of the arising subcategories is reflective. We give an iterative construction of the reflectors and present characteristic examples.
We introduce the concept of firm classes of morphisms as basis for the axiomatic study of completions of objects in arbitrary categories. Results on objects injective with respect to given morphism classes are included. In a finitely well-complete category, firm classes are precisely the coessential first factors of morphism factorization structures.
The contribution is devoted to the question of the interchange of the construction of a quasiorder hypergroup from a quasiordered set and the factorization.