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  "body": "\n## Definition\n\nLet $B$ be a topological space and let $p \\colon E \\to B$ and $p' \\colon E' \\to B$ be real (or complex) vector bundles over $B$, of ranks $n$ and $n'$.\nThe *Whitney sum* of $E$ and $E'$ is the space\n$$E \\oplus E' = \\set{(v, v') \\in E \\times E' : p(v) = p'(v')},$$\nwith the [[subspace topology|0WS9P12FQR]] of the [[product|0GBG8B49AA]] $E \\times E'$, together with the map $q \\colon E \\oplus E' \\to B$, $q(v, v') = p(v)$.\nThe fiber $q^{-1}(b) = E_b \\times E'_b$ is given the [[vector space|000AYPA9AJ]] structure of the direct sum $E_b \\oplus E'_b$.\n\n## Remarks\n\nThe map $q$ is a vector bundle of rank $n + n'$: local trivializations $p^{-1}(U) \\cong U \\times \\R^n$ and $p'^{-1}(U) \\cong U \\times \\R^{n'}$ combine into $q^{-1}(U) \\cong U \\times \\R^{n + n'}$.\nEquivalently, $E \\oplus E'$ is the pullback of the product bundle $E \\times E' \\to B \\times B$ along the diagonal $B \\to B \\times B$.\nThe Whitney sum is associative and commutative up to isomorphism, and the trivial bundle of rank $0$ is a unit.\n\nExample: let $TS^2$ be the tangent bundle of $S^2 \\subset \\R^3$ and $\\nu = \\set{(x, t x) : x \\in S^2, t \\in \\R}$ the normal line bundle.\nAt each $x \\in S^2$, $T_x S^2 \\oplus \\R x = \\R^3$, and so $TS^2 \\oplus \\nu$ is isomorphic to the trivial bundle $S^2 \\times \\R^3$, although $TS^2$ itself is not trivial.\n\nNon-example: for bundles $E \\to B$ and $E' \\to B'$ over different bases, the product $E \\times E' \\to B \\times B'$ is a vector bundle of rank $n + n'$.\nIt is not a Whitney sum, which needs a common base; it is the external direct sum.\n",
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