Covering groups of non-connected topological groups revisited

Abstract

All spaces are assumed to be locally path connected and semi-locally 1-connected. Let X be a connected topological group with identity e , and let be the universal cover of the underlying space of X . It follows easily from classical properties of lifting maps to covering spaces that for any point ẽ in with p ẽ = e there is a unique structure of topological group on such that ẽ is the identity and is a morphism of groups. We say that the structure of topological group on X lifts to .