For a covering space p:X~→X with X connected, locally path-connected, and semilocally simply-connected, show:
((a)) The components of X~ are in one-to-one correspondence with the orbits of the action of π1(X,x0) on the fiber p−1(x0).
((b)) Under the Galois correspondence between connected covering spaces of X and subgroups of π1(X,x0), the subgroup corresponding to the component of X~ containing a given lift x~0 of x0 is the stabilizer of x~0, the subgroup consisting of elements whose action on the fiber leaves x~0 fixed.