Existence of a simply connected covering space:

Despite being an important theme, we shall not discuss this in any detail in this elementary course but make a few remarks about it. Most of the spaces that we shall encounter are reasonably well-behaved and indeed many of them such $SO(n, \mathbb{R})$, $S^3$ and the projective spaces are smooth manifolds. Given the existence of a simply connected covering - called a universal covering⁴, one can develop a Galois correspondence for covering spaces which asserts the existence of a unique (upto isomorphism) covering corresponding to each conjugacy class of subgroups of $\pi_1(X, x_0)$.