Question in the appendixes A of O'Neill book

I'm reading appendixes A of O'Neill's book, "Semi-riemannian geometry" and I don't understand a something.

I don't understand at the last theorem, how we construct a universal cover for $M$ and why is that (if it is) a manifold.

• The construction is not described in the paragraph you quote. He only claims that it is legit to discuss the simply connected covering, as opposed to a simply connected covering. – Amitai Yuval Aug 14 '18 at 17:52
• Yeah, sure but i put the whole paragraph to show the lack of explanation and as a reference for someone who might use this as a tool for the proof. So my question becomes where can I find a proof for this theorem? – Hurjui Ionut Aug 14 '18 at 18:55
• If you’re willing to invest some time it’s in chapter 3 of Peter May book math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf – Elad Aug 15 '18 at 8:58