 Monodromy action of a covering space

Let $p:E\to X$ be a covering space with path-connected base, and fix a base point $x_0\in X$. Lift a based loop $\alpha$ from each $e\in p^{-1}(x_0)$ and let $e\cdot [\alpha ]$ be its endpoint. The path lifting theorem and homotopy lifting property make this a well-defined permutation of the fibre. With loop multiplication $\alpha \ast \beta$ meaning traverse $\alpha$ first and then $\beta$, it is a right group action:Its orbits correspond exactly to the path components of $E$. Indeed, two fibre points are joined by a path in $E$ precisely when projecting such a path gives a based loop whose lift joins them. Every path component meets the fibre, since a path from any base point to $x_0$ can be lifted.

Consequently this monodromy action of a covering space is transitive if and only if $E$ is path-connected. If $X$ is also locally path-connected, then $E$ is locally path-connected too, and connectedness of $E$ suffices. The local condition explains the familiar formulation using a connected cover.