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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, Hausdorff and second countable.\nA *smooth embedding* is an [[immersion|6RW68JGBHC]] that is a homeomorphism onto its image.\nThen there is a smooth embedding of $M$ into the [[Euclidean space|0XPR1E85W0]] $\\R^{2n}$.\nIn particular $M$ is diffeomorphic to an [[embedded submanifold|VKFW27WXEZ]] of $\\R^{2n}$.\n\n## Remarks\n\nThere is also a weak version: every such $M$ embeds in $\\R^{2n+1}$, and any smooth map $M \\to \\R^{2n+1}$ can be approximated by embeddings.\nIts proof starts from an embedding into some large $\\R^N$, built from charts and bump functions.\nThen $N$ is lowered by generic linear projections.\nSard's theorem shows that almost every projection stays an injective immersion while $N \u003e 2n + 1$.\nTo reach $2n$, one first takes an immersion into $\\R^{2n}$ with only transverse double points.\nThen one removes the double points in pairs; for $n \\ge 3$ this step uses the Whitney trick, and $n = 1, 2$ are treated directly.\nThe bound $2n$ is sharp in general: for $n$ a power of $2$, the real projective space $\\R P^n$ does not embed in $\\R^{2n - 1}$.\nExample: the circle ($n = 1$) embeds in $\\R^2$, and the Klein bottle ($n = 2$) embeds in $\\R^4$ but not in $\\R^3$.\n\n## Explanations\n\n### L1 human\n\nEvery smooth manifold, however it was defined, can be realised as a smooth shape inside an ordinary Euclidean space.\nTwice its dimension is always enough room.\nSo abstract manifolds are no more general than the curved surfaces and solids that sit inside some $\\R^N$.\n\n### L2 human\n\nThe theorem reconciles the intrinsic definition of a manifold by charts with the older picture of a manifold as a subset of $\\R^N$.\nIt allows one to borrow structures from $\\R^N$, such as a Riemannian metric induced from the dot product or a normal bundle and a tubular neighbourhood.\nThe weak version is a typical general position argument, a precursor of transversality theory.\nThe strong version introduced the Whitney trick, which later became the key step in the h-cobordism theorem.\n",
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