What has not been said is that if $X$ is path connected and has a universal cover, then the fundamental groupoid $\pi_1(X)$ can be given a topology so that the source map $s:\pi_1 (X) \to X$ is a bundle over $X$ with fibre over $x \in X$ the universal cover of $X$ based at $x$: see Chapter 10 of Topology and Groupoids. This has been known for a long time.
Ronnie Brown
- 12.3k
- 1
- 63
- 81