On Generalized Covering Groups of Topological Groups

It is well-known that a homomorphism p between topological groups K, G is a covering homomorphism if and only if p is an open epimorphism with discrete kernel. In this paper we generalize this fact, in precisely, we show that for a connected locally path connected topological group G, a continuous map p is a generalized covering if and only if K is a topological group and p is an open epimorphism with prodiscrete (i.e, product of discrete groups) kernel. To do this we first show that if G is a topological group and H is any generalized covering subgroup of fundamental group of G, then H is as intersection of all covering subgroups, which contain H. Finally, we show that every generalized covering of a connected locally path connected topological group is a fibration.