Let $n \in \mathbb{N}$, let $E$ and $B$ be smooth manifolds, and let $p: E \to B$ be an $n$-sheeted smooth covering map. That is, for every $b \in B$ there exists an open neighbourhood $U \subset B$ of $b$ such that
where each $V_i \subset E$ is open and the restricted map $p|_{V_i}: V_i \to U$ is a diffeomorphism.
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Then $p: E \to B$ is a smooth fibre bundle with fibre $F := \{1,\dots,n\}$, where $F$ is equipped with the discrete smooth structure. Moreover, the bundle admits structure group $S_n$, acting on $F$ by permutations.