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  "body": "\n## Example\n\nLet $n \\ge 1$.\nFor $k \\ge n$, let $G_n(\\R^k)$ be the Grassmannian of $n$-dimensional linear subspaces of $\\R^k$.\nThe inclusions $\\R^k \\subset \\R^{k+1}$ give inclusions $G_n(\\R^k) \\subset G_n(\\R^{k+1})$, and $G_n(\\R^\\infty) = \\bigcup_k G_n(\\R^k)$ carries the weak topology of this union.\nThe tautological bundle\n$$\\gamma^n = \\set{(V, v) \\in G_n(\\R^\\infty) \\times \\R^\\infty : v \\in V}$$\nis a real vector bundle of rank $n$ over $G_n(\\R^\\infty)$.\n\nLet $X$ be a paracompact space.\nThen $[f] \\mapsto f^* \\gamma^n$ is a bijection from homotopy classes of maps $X \\to G_n(\\R^\\infty)$ to isomorphism classes of real rank $n$ bundles over $X$.\nRank $n$ real vector bundles correspond to principal $O(n)$-bundles through their bundles of orthonormal frames.\nSo $G_n(\\R^\\infty)$ is a [[classifying space|WSZ40H5E4T]] $BO(n)$ of the orthogonal group $O(n)$.\nThe frame bundle of $\\gamma^n$ is the universal bundle; its total space is the infinite [[Stiefel manifold|W7RCXS3CHT]] $V_n(\\R^\\infty)$, which is contractible.\n\n## Remarks\n\nFor $n = 1$, $G_1(\\R^\\infty) = \\R P^\\infty$, and real line bundles over a CW complex $X$ correspond to classes in $H^1(X; \\Z/2\\Z)$ through the first [[Stiefel–Whitney class|JQCK7A210W]].\nIn the same way, the complex Grassmannian $G_n(\\C^\\infty)$ is $BU(n)$, and $\\C P^\\infty = BU(1)$ classifies complex line bundles.\nThe cohomology ring $H^*(G_n(\\R^\\infty); \\Z/2\\Z)$ is the polynomial ring over $\\Z/2\\Z$ on $w_1(\\gamma^n), \\ldots, w_n(\\gamma^n)$, which is why every mod $2$ [[characteristic class|7HACJC0JE2]] of rank $n$ bundles is a polynomial in Stiefel–Whitney classes.\n\n## Explanations\n\n### L1 human\n\nA rank $n$ vector bundle over $X$ attaches an $n$-dimensional space of vectors to each point of $X$.\nPlace every such space inside one huge space $\\R^\\infty$.\nThen the bundle becomes a map that sends each point of $X$ to an $n$-dimensional subspace, that is, to a point of the Grassmannian.\nConversely, any map into the Grassmannian gives a bundle: over each point of $X$, take the subspace it is sent to.\nSo the infinite Grassmannian lists all vector bundles of rank $n$ at once.\n",
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