Let $B$ and $F$ be nonempty smooth manifolds, and let $((U_i)_{i\in I},(\gamma_{ij}))$ be a smooth bundle atlas covering $B$ with fibre $F$. Suppose the quotient of $\bigsqcup_i U_i\times F$ by the identifications prescribed by $\gamma_{ij}$ is Hausdorff and second-countable with the [quotient topology](/page/Quotient%20Topology) induced by the local charts. Then there exists a smooth fibre bundle $\pi:E\to B$ with fibre $F$ and local trivializations $\Phi_i:\pi^{-1}(U_i)\to U_i\times F$ whose changes of trivialization are the maps $\gamma_{ij}$. The bundle is unique up to a diffeomorphism over $B$ preserving the given local trivializations.