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A fiber bundle \((E,B,p,F)\) is a bundle of topological spaces \((E,B,p)\) along with a third topological space \(F\) (the 'fiber') such that \(\forall x\in B:p^{-1}(B)\cong F\), i.e. the preimage of each point of the base-space under projection is homeomorphic to the fiber, and also for each \(x\in B\), \(\exists U\in\mathbf{Top}(B):\exists(\phi:p^{-1}(U)\to U\times F)\in\mathbf{Top}:p=\pi_U\circ\phi\) - there is an open neighbourhood \(U\) of every point \(x\) of the base space, upon the preimage of which the projection factors through the projection onto \(U\) from the trivial product \(U\times F\).