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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 map|0X3WGX95FE]].\nLet $U$ be an open subset of $B$, and give $U \\times F$ the [[product topology|0GBG8B49AA]].\nA *local trivialization* of $\\pi$ over $U$ with fiber $F$ is 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.\n\n## Remarks\n\nExample: let $E = \\R \\times S^1$, $B = S^1$ and $\\pi(s, z) = z$.\nThen $\\varphi(s, z) = (z, s)$ over $U = S^1$ is a local trivialization with fiber $\\R$, since it is a homeomorphism and its first coordinate is $\\pi(s, z) = z$.\n\nNon-example: for the double cover $\\pi \\colon S^1 \\to S^1$, $\\pi(z) = z^2$, there is no local trivialization over $U = S^1$ with fiber a two-point space, since $\\pi^{-1}(S^1) = S^1$ is connected and $S^1 \\times F$ is not.\nOver each proper open arc, however, such a trivialization exists.\n\nThe map $\\pi$ is a [[fiber bundle|CX4M8K61AB]] with fiber $F$ exactly when the open sets carrying a local trivialization cover $B$.\nIf $\\varphi_U$ and $\\varphi_V$ are local trivializations, then $\\varphi_U \\circ \\varphi_V^{-1}$ on $(U \\cap V) \\times F$ has the form $(b, f) \\mapsto (b, g_{UV}(b)(f))$, and the maps $g_{UV}$ are the transition functions.\n",
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