On Continuity and Openness of Homomorphisms in Topological Groups

Abstract

In the present paper we wish to give certain necessary and sufficient conditions that a homomorphism h between two topological groups X and Y be continuous, and similar conditions that it be open, i.e., map open sets into open sets. The present results may be considered as extensions to less restricted groups of theorems known to hold in Banach spaces or in spaces of Type (F) or in separable groups. The guiding principle has been that of transferring as much as possible the restrictions in these earlier theorems from the groups to the homomorphism. In Part I, after extending Banach's theorem on second category Baire subgroups to show (Theorem 1) that the identity element always lies in the of A_'A whenever A contains a second category Baire set, it is proved in Theorem 2 that h is continuous and X second category if and only if h-'(V) is either null or contains a second category Baire set whenever V is open in Y; this is easily dualized to show that h is open and Y second category if and only if h(U) contains a second category Baire set for each non-null open U in X. From these there follow theorems given by Banach [2, 3], Bourbaki [6], Freudenthal [9], and McShane [15]. There are also extensions to linear topological spaces of certain results of Banach [2] and of Mazur and Orlicz [14] for Banach spaces and spaces of Type (F). In Part II our methods, based on a proof of Banach's, require that X and Y satisfy the first countability axiom. The essential result here (Theorem 15) is that if X is right complete, i.e., complete with respect to at least one of its rightinvariant metrics, then h is open and has a kernel if and only if it has a and maps non-null open sets into somewhere dense sets. The dual of this also holds (Corollary 16.1); if Y is right complete and is a Hausdorff group then h is continuous if and only if it has a and h-' maps each open set in Y into either the null set or a somewhere dense set. These are extensions of Banach's interior mapping and closed graph theorems in Type (F) spaces [2]. A theorem of Dunford's [8] also follows with weaker hypotheses. In conclusion there are indications of some extensions of the present results. By a topological space (TS) we shall understand as usual a space X with a distinguished class of subsets, called open sets, which is under arbitrary