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A bundle \(\pi\colon E\to X\) is further called a fibre bundle with fibre \(F\) (for \(F\) a manifold) if, for each element \(x\) of the base space its preimage \(\pi^{-1}(x)\) is diffeomorphic to \(F\), and such that \(\pi\colon E\to X\) is locally equivalent to the trivial bundle \(F\times X\), i.e. one may find a covering \((U_i)_{i\in I}\) of \(X\) such that for each open set \(U_i\) there is a diffeomorphism \(\phi_i:\pi^{-1}(U_i)\to U_i\times F\) such that \(\pi=\pi_{U_i}\circ\phi_i\) (where \(\pi_{U_i}\) is projection onto the first factor). Alternatively, one may construct a fibre bundle with fibre \(F\) and base space \(X\) by taking a Lie group \(G\), an injection \(\rho\colon G\to\mathrm{Aut}(F)\), a cover \((U_i)_{i\in I}\) of \(X\) and a cocycle of the cover with values in \(G\) (in the sense of Cech cohomology), i.e. for each \(i,j\in I\) a smooth map \(g_{ij}\colon U_i\cap U_j\to G\) such that \(g_{ii}(x)=1_G\), \(g_{ij}(x)g_{ji}(x)=1_G\) and, for all \(i,j,k\in I,\:x\in U_i\cap U_j\cap U_k\), \(g_{jk}(x)g_{ij}(x)=g_{ik}(x)\). Then one may take the disjoint union \(\sqcup_{i\in I}U_i\times F\) and quotient by the relation \((x,f)\sim (x,\rho(g_{ij}(x))(f)\) whenever \(x\in U_i\cap U_j\) to form a fibre bundle.