We introduce a relativized notion of (semi)normalcy for categories that come equipped with a proper stable factorization system, and we use radicals (in the sense of module theory) and normal
closure operators in order to study torsion theories in such categories. Our results generalize and complement
recent studies in the realm of semi-abelian and, in part, homological categories. In particular, we characterize both, torsion-free and torsion classes, in terms of their closure under extensions. We pay particular attention to the homological and, for our purposes more importantly, normal categories
of topological algebra, such as the category of topological groups. But our applications go far beyond
the realm of these types of categories, as they include, for example, the normal, but non-homological category of pointed topological spaces, which is in fact a rich supplier for radicals of topological groups.
