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  "body": "\n## Description\n\nLet $n \\ge 0$ be an integer.\nThe *$n$-sphere* is the subspace\n$$S^n = \\set{x \\in \\R^{n+1} : x_1^2 + \\cdots + x_{n+1}^2 = 1}$$\nof [[Euclidean space|0XPR1E85W0]] $\\R^{n+1}$, the set of points at distance $1$ from the origin.\nIt is a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.\nIt is [[Hausdorff|0S1FJ8VEX9]] and [[second-countable|03PPNCM9GG]] as a subspace of $\\R^{n+1}$.\nFor the north pole $N = (0, \\ldots, 0, 1)$, the stereographic projection\n$$\\sigma_N(x) = \\frac{(x_1, \\ldots, x_n)}{1 - x_{n+1}}$$\nis a [[homeomorphism|0QT00PB5G2]] from $S^n \\setminus \\set{N}$ onto $\\R^n$, with inverse $y \\mapsto \\bigl(2y, \\abs{y}^2 - 1\\bigr) / \\bigl(\\abs{y}^2 + 1\\bigr)$.\nThe projection from the south pole is defined in the same way, and the two charts cover $S^n$.\n\n## Remarks\n\n$S^0$ is a pair of points, $S^1$ is the [[circle|060W5CBZP6]] and $S^2$ is the usual round sphere.\n$S^n$ is [[compact|0TVJZYB6SH]], being closed and bounded in $\\R^{n+1}$, and for $n \\ge 1$ it is [[path-connected|0WB6DH499T]].\nFor $n \\ge 2$ it is simply connected.\nFor $n \\ge 1$ it is the one-point compactification of $\\R^n$, and it is homeomorphic to the closed ball $D^n$ with its boundary sphere collapsed to a point.\nIts integer homology is $\\Z$ in degrees $0$ and $n$ and zero in the other degrees, for $n \\ge 1$.\n\n## Explanations\n\n### L1 human\n\nThe $n$-sphere is the set of points at distance one from a centre, in one dimension more than the sphere itself.\nThe $1$-sphere is a circle in the plane, the $2$-sphere is the surface of a ball in space.\nNear each point it looks like flat $n$-dimensional space, but as a whole it closes up on itself.\n",
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