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  "body": "\n## Definition\n\nLet $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] and let $C^\\infty(M, N)$ be the set of smooth maps from $M$ to $N$.\nFor an integer $k \\ge 0$, let $J^k(M, N)$ be the manifold of [[$k$-jets|1QE0Z0756D]] of maps from $M$ to $N$, the [[jet bundle|HC3VBBCJQP]] of the trivial bundle $M \\times N \\to M$.\nEach $f \\in C^\\infty(M, N)$ has the jet map $j^k f \\colon M \\to J^k(M, N)$, $p \\mapsto j^k_p f$.\nFor an open set $U \\subseteq J^k(M, N)$, put\n$$\\mathcal{M}(U) = \\set{ f \\in C^\\infty(M, N) : j^k f(M) \\subseteq U }.$$\nThe sets $\\mathcal{M}(U)$, with $U$ open in $J^k(M, N)$, form a base of a [[topology|0CQHFPGJ45]] on $C^\\infty(M, N)$, the *Whitney $C^k$ topology*.\nThe *Whitney $C^\\infty$ topology* is the topology whose base is the union over all $k \\ge 0$ of these bases.\n\n## Remarks\n\nWith the Whitney $C^\\infty$ topology, $C^\\infty(M, N)$ is a [[Baire space|0ZH5AJ2ED7]].\nIf $M$ is compact, it coincides with the weak $C^\\infty$ topology, the topology of uniform convergence of all derivatives, which is built from the [[compact-open topology|RG15F0TWPW]] on jets.\nIf $M$ is not compact, it is much finer.\n\nExample: for $M = N = \\R$, let $\\varepsilon \\colon \\R \\to (0, \\infty)$ be continuous and let $f \\in C^\\infty(\\R, \\R)$.\nThe set $U = \\set{ (x, y) \\in \\R \\times \\R : \\abs{y - f(x)} \u003c \\varepsilon(x) }$ is open in $J^0(\\R, \\R) = \\R \\times \\R$.\nSo $\\mathcal{M}(U) = \\set{ g : \\abs{g(x) - f(x)} \u003c \\varepsilon(x) \\text{ for all } x }$ is an open neighbourhood of $f$.\n\nNon-example: the constant maps $f_n = 1/n$ do not converge to $0$ in the Whitney $C^\\infty$ topology on $C^\\infty(\\R, \\R)$, although they converge uniformly with all derivatives.\nTake $f = 0$ and $\\varepsilon(x) = 1/(1 + x^2)$ above.\nFor each $n$, $1/n \u003e \\varepsilon(x)$ for large $\\abs{x}$, so no $f_n$ lies in this neighbourhood of $0$.\n",
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