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  "body": "\n## Definition\n\nA *projective geometry*, or projective space in the axiomatic sense, is an [[incidence structure|FGKNAFNHA3]] whose lines are sets of points, incidence being membership, such that:\n\n1. Any two distinct points lie on exactly one line.\n2. (Veblen's axiom) Let $a, b, c, d$ be distinct points; if the lines $ab$ and $cd$ meet, then the lines $ac$ and $bd$ meet.\n3. Every line contains at least three points.\n\nThe subject *projective geometry* is the study of these spaces and of the properties preserved by their isomorphisms.\n\n## Remarks\n\nExample: let $V$ be a vector space of dimension at least $3$ over a field $k$.\nThe points of the [[projective space|ZT9A6JK8QF]] $\\mathbb{P}(V)$, with the lines $\\mathbb{P}(W)$ for $\\dim W = 2$, form a projective geometry.\nCheck: two distinct points span a unique $2$-dimensional $W$.\nIf $ab$ and $cd$ meet, then $a, b, c, d$ lie in a subspace $U$ of dimension at most $3$.\nThe planes for $ac$ and $bd$ lie in $U$, so they meet in a nonzero vector.\nA $2$-dimensional $W$ has $\\abs{k} + 1 \\ge 3$ points.\n\nNon-example: the real [[affine plane|EC7XT2ZM5S]] $\\R^2$ satisfies clauses 1 and 3 but fails Veblen's axiom.\nTake $a = (0,0)$, $b = (1,0)$, $c = (0,1)$ and $d = (1,2)$.\nThe lines $ab$ (given by $y = 0$) and $cd$ (given by $y = x + 1$) meet at $(-1, 0)$, but the lines $ac$ (given by $x = 0$) and $bd$ (given by $x = 1$) are parallel.\n\nA projective geometry of dimension $2$ is the same as a [[projective plane|DANFNN8C4Y]].\nBy the [[Veblen–Young theorem|NPNNXWMVCC]], every projective geometry of dimension at least $3$ is some $\\mathbb{P}(V)$ over a division ring.\n",
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