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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $0 \\le k \\le n$.\nA subset $S \\subseteq M$ is an *embedded submanifold* of dimension $k$ if the following holds.\nFor every point $p \\in S$ there is a chart $(U, \\varphi)$ of the smooth [[atlas|J7P32JQTVE]] of $M$ with $p \\in U$ such that\n$$\\varphi(U \\cap S) = \\varphi(U) \\cap (\\R^k \\times \\set{0}).$$\nSuch a chart is a *slice chart* for $S$.\nThe restrictions of the slice charts give $S$ a unique smooth structure for which the inclusion $S \\to M$ is a smooth embedding.\n\n## Remarks\n\nExample: the unit circle $S^1 \\subseteq \\R^2$ is an embedded submanifold of dimension $1$.\nNear a point with $y \u003e 0$, the chart $(x, y) \\mapsto (x, y - \\sqrt{1 - x^2})$ on $\\set{(x, y) : \\abs{x} \u003c 1, y \u003e 0}$ sends the circle onto the axis $\\R \\times \\set{0}$; the other points are handled in the same way.\nNon-example: the cross $C = \\set{(x, y) \\in \\R^2 : xy = 0}$ is not an embedded submanifold.\nRemoving the origin from a small connected neighbourhood of $0$ in $C$ leaves four components.\nRemoving a point from an open interval leaves two, so no slice chart exists at the origin.\nThe embedded submanifolds of $M$ are exactly the images of smooth embeddings into $M$.\nAn *immersed submanifold* is the image of an injective [[immersion|6RW68JGBHC]] $S \\to M$, with the topology and smooth structure carried over from $S$; its topology may be finer than the subspace topology.\n",
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