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  "body": "\n## Definition\n\nLet $E$, $B$ and $F$ be [[topological spaces|0CQHFPGJ45]], and let $\\pi \\colon E \\to B$ be a [[continuous|0X3WGX95FE]] surjection.\nThe quadruple $(E, B, \\pi, F)$ is a *fiber bundle* if the following holds.\nEvery point $b \\in B$ has an open neighbourhood $U$ and a [[homeomorphism|0QT00PB5G2]]\n$$\\varphi \\colon \\pi^{-1}(U) \\to U \\times F$$\nsuch that $\\operatorname{pr}_1 \\circ \\varphi = \\pi$ on $\\pi^{-1}(U)$, where $\\operatorname{pr}_1 \\colon U \\times F \\to U$ is the first projection.\nSuch a pair $(U, \\varphi)$ is a *local trivialization*.\nThe spaces $E$, $B$ and $F$ are the total space, the base and the fiber, and $\\pi$ is the projection.\n\n## Remarks\n\nFor any spaces $B$ and $F$, the product $B \\times F$ with $\\pi = \\operatorname{pr}_1$ is a fiber bundle, the trivial bundle; one trivialization over $U = B$ suffices.\nThe Möbius band is a fiber bundle over the circle with fiber $[0, 1]$ that is not trivial.\n\nNon-example: let $E = \\set{(x, y) \\in \\R^2 : xy = 0}$ and $\\pi(x, y) = x$, so $\\pi \\colon E \\to \\R$.\nThe preimage of $0$ is a line, while the preimage of each $x \\neq 0$ is a single point.\nA local trivialization around $0$ would make all preimages near $0$ homeomorphic to the same $F$, which is impossible.\n\nEach fiber $\\pi^{-1}(b)$ is homeomorphic to $F$.\nA *smooth fiber bundle* uses smooth manifolds, a smooth submersion $\\pi$ and diffeomorphisms $\\varphi$.\nThe notation $F \\to E \\to B$ is common.\n\n## Explanations\n\n### L1 human\n\nA fiber bundle is a map onto a base space that looks like the projection of a product over small open sets of the base.\n",
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