Functional topologies on Abelian groups

Abstract

introduced the concept of "functorial topology" which includes the ubiquitous /7-adic and Z-adic topologies. He proposed a method for constructing such topologies which was slightly generalized by Fuchs [4], Vol I, p. 33. Closer inspection shows that this method amounts to specifying the class of discrete groups and furnishing all other groups with the coarsest topology required to make all homomorphisms continuous. The "discrete classes" satisfy certain closure properties and studying these classes enables us to partially solve Fuchs' Problem 2, [4], asking for a description of functorial linear topologies. There is no bijective correspondence between functorial topologies and discrete classes -examples are readily available (see 2.8) -but there is a bijective correspondence between the "minimal functorial topologies" obtained via the Charles-Fuchs construction and discrete classes (Theorem 2.5). We call a functorial topology "ideal" if in addition to having continuous homomorphisms all epimorphisms are open maps. This is true for the /?-adic and Z-adic topologies, for example. We obtain a bijective correspondence between linear ideal functorial topologies and "ideal" discrete classes. The latter are satisfactorily characterized (3.5, 3.11, 3.19) except for one nasty case.