On the classes of dense and closed subobjects (Q1916419)

Let \({\mathcal X}\) be a category that is \({\mathcal M}\)-complete with respect to a class \({\mathcal M}\) of monomorphisms containing all isomorphisms and closed under composition (i.e., \({\mathcal M}\)-subobjects admit arbitrary intersections and are stable under pullbacks). The author attempts to characterize the classes of \({\mathcal M}\)-subobjects that can arise as classes of dense \({\mathcal M}\)-subobjects for a closure operator on \({\mathcal M}\) by three conditions, dualizing a known result for the classes of closed \({\mathcal M}\)-subobjects. Since \(m\in{\mathcal M}\) is dense iff \(\langle m,id\rangle\) is relatively dense, the author's result is an easy consequence of theorem 1 by \textit{G. Castellini}, \textit{G. E. Strecker} and the reviewer [in: Proceedings of the Seventh Summer Conference at the University of Wisconsin 1991, Ann. N. Y. Acad. Sci. 704, 38-52 (1993; Zbl 0815.18001)], which characterizes classes of relatively dense \({\mathcal M}\)-pairs. The author applies his characterization to answer a question posed by D. Dikranjan. In the category of topological abelian groups with continuous homomorphisms the class of totally dense subgroups (i.e., subgroups that, topologically, have a dense intersection with every closed subgroup with the same codomain) is not the class of dense subgroups for any closure operator.