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  "body": "\n## Definition\n\nLet $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], and give the set $C^\\infty(M, N)$ of smooth maps the [[Whitney $C^\\infty$ topology|7PN9MD1PQB]].\nTwo smooth maps $f, g \\colon M \\to N$ are *equivalent* if there are [[diffeomorphisms|0KZZ70ZHXP]] $\\varphi \\colon M \\to M$ and $\\psi \\colon N \\to N$ with $g = \\psi \\circ f \\circ \\varphi^{-1}$.\nA smooth map $f \\colon M \\to N$ is *stable* if there is a neighbourhood $U$ of $f$ in $C^\\infty(M, N)$ such that every $g \\in U$ is equivalent to $f$.\n\n## Remarks\n\nExample: the height function $f \\colon S^1 \\to \\R$, $f(x, y) = y$, on the unit circle is stable.\nIts critical points $(0, \\pm 1)$ are nondegenerate with distinct values $\\pm 1$.\nAny $g$ close enough to $f$ in the $C^2$ sense also has exactly two nondegenerate critical points, a maximum and a minimum.\nTheir values are distinct, and such a $g$ is equivalent to $f$.\nNon-example: a constant map $c \\colon S^1 \\to \\R$ is not stable.\nThe maps $c + \\varepsilon f$ come arbitrarily close to $c$ and have nonzero differential at some points, while every map equivalent to $c$ is constant.\nFor a compact manifold $M$, a function $f \\colon M \\to \\R$ is stable if and only if all its critical points are nondegenerate and its critical values are pairwise distinct.\nThe equivalence used here is often called $\\mathcal{A}$-equivalence, or left-right equivalence.\n",
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