Let $B$ and $F$ be smooth manifolds, let $E$ be a set, and let $\pi:E\to B$ be a surjective map. Suppose that $(U_i)_{i\in I}$ is an open cover of $B$ and that, for each $i\in I$, there is a bijection
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\begin{align*}
\Phi_i:\pi^{-1}(U_i) &\to U_i\times F
\end{align*}
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such that $\operatorname{pr}_1\circ \Phi_i=\pi|_{\pi^{-1}(U_i)}$, where $\operatorname{pr}_1:U_i\times F\to U_i$ is the first projection.
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Give $E$ the topology in which a subset $W\subset E$ is open if and only if, for every $i\in I$, the set
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\begin{align*}
\Phi_i\left(W\cap \pi^{-1}(U_i)\right)\subset U_i\times F
\end{align*}
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is open. Assume that this topology is Hausdorff and second-countable. Suppose moreover that, whenever $U_i\cap U_j\neq \varnothing$, the transition map
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\begin{align*}
\Phi_i\circ \Phi_j^{-1}:(U_i\cap U_j)\times F &\to (U_i\cap U_j)\times F
\end{align*}
for some family of diffeomorphisms $g_{ij}(b):F\to F$ depending smoothly on $b\in U_i\cap U_j$ in the sense that the displayed transition map is smooth.
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Then there is a unique smooth manifold structure on the [topological space](/page/Topological%20Space) $E$ such that each map
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\begin{align*}
\Phi_i:\pi^{-1}(U_i)&\to U_i\times F
\end{align*}
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is a diffeomorphism. With this smooth structure, $\pi:E\to B$ is a smooth fibre bundle over $B$ with typical fibre $F$ and local trivializations $(\Phi_i)_{i\in I}$.