# Existence of the universal covering space of a connected Lie group

I am working on a project about how the universal cover of a connected Lie group is a Lie group, but I cannot find a theorem that assures that this universal cover actually exists. I've found references on:

1. Lie Groups, An Approach through Invariants and Representations, by Claudio Procesi: page 80.

2. Probability on Compact Lie Groups by David Applebaum: page 5.

But I have not found the actual statement nor proof of the result anywhere. Does anybody know where I may find it?

• I'm sure you've figured it out by now, but if you know the fact for topological groups, checking smoothness of multiplication and inversion should be the least "nightmarish" part because the covering space gets its smooth structure from the space it's covering. The covering map is a local diffeomorphis, by definition, and smoothness is a local property. – Tim kinsella Jun 8 '15 at 22:09