= 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 $\alpha*\beta$ meaning traverse $\alpha$ first and then $\beta$, it is a right <group action>:
$$
(e\cdot[\alpha])\cdot[\beta]=e\cdot[\alpha*\beta].
$$
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.
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