Closure operators and functorial topologies

Abstract

A Galois connection between subcategories of a given category A and global closure operators on A leads to a Galois connection between subcategories of abelian groups and functorial topologies. In particular each functorial topology F τ determines a global closure operator F, whose subcategory of separated objects, D( F), is the class of Hausdorff groups of F τ. Given a subcategory C of abelian groups, one can always construct a functorial topology whose Hausdorff groups form the epireflective hull of C . It is also shown that in certain concrete categories, the largest subcategory that induces the same closure operator as a given subcategory C , is the extremal-epireflective hull of C .