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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $n \\ge 0$ be an integer.\nOn $k^{n+1} \\setminus \\set{0}$, let $x \\sim y$ when $y = \\lambda x$ for some $\\lambda \\in k \\setminus \\set{0}$; this is an [[equivalence relation|0G8VEZ3WTK]].\nThe *projective $n$-space* over $k$ is the set of equivalence classes\n$$\\mathbb{P}^n = \\mathbb{P}^n(k) = \\left(k^{n+1} \\setminus \\set{0}\\right) / \\sim.$$\nThe class of $(x_0, \\dots, x_n)$ is written $[x_0 : \\dots : x_n]$, and the $x_i$ are its homogeneous coordinates.\n\n## Remarks\n\nEquivalently, $\\mathbb{P}^n$ is the set of one-dimensional [[linear subspaces|00YC1D652K]] of the [[vector space|000AYPA9AJ]] $k^{n+1}$.\nMore generally, $\\mathbb{P}(V)$ denotes the set of one-dimensional subspaces of a vector space $V$.\n\nExample: $[1 : 2 : 3] = [2 : 4 : 6]$ in $\\mathbb{P}^2(\\Q)$, because $(2, 4, 6) = 2 \\cdot (1, 2, 3)$.\nOver $\\R$ with its usual topology, $\\mathbb{P}^n(\\R)$ is the [[real projective space|1DDYB6VVSR]].\n\nNon-example: $(0 : 0 : 0)$ is not a point of $\\mathbb{P}^2$, since the zero vector is excluded.\n\nFor each $i$, the set $U_i = \\set{x_i \\ne 0}$ is in bijection with $k^n$ through $[x_0 : \\dots : x_n] \\mapsto (x_0/x_i, \\dots, \\widehat{x_i/x_i}, \\dots, x_n/x_i)$, and the $U_i$ cover $\\mathbb{P}^n$.\nThe complement of $U_0$ is $\\set{x_0 = 0}$, a copy of $\\mathbb{P}^{n-1}$, the hyperplane at infinity.\n",
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