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Finite geometry
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===Axiomatic definition=== A '''projective space''' ''S'' can be defined axiomatically as a set ''P'' (the set of points), together with a set ''L'' of subsets of ''P'' (the set of lines), satisfying these axioms :<ref>{{harvnb|Beutelspacher|Rosenbaum|1998|loc=pp. 6–7}}</ref> * Each two distinct points ''p'' and ''q'' are in exactly one line. * [[Oswald Veblen|Veblen]]'s axiom:<ref>also referred to as the ''Veblen–Young axiom'' and mistakenly as the [[Pasch's axiom|axiom of Pasch]] {{harv|Beutelspacher|Rosenbaum|1998|loc=pgs. 6–7}}. Pasch was concerned with real projective space and was attempting to introduce order, which is not a concern of the Veblen–Young axiom.</ref> If ''a'', ''b'', ''c'', ''d'' are distinct points and the lines through ''ab'' and ''cd'' meet, then so do the lines through ''ac'' and ''bd''. * Any line has at least 3 points on it. The last axiom eliminates reducible cases that can be written as a disjoint union of projective spaces together with 2-point lines joining any two points in distinct projective spaces. More abstractly, it can be defined as an [[incidence structure]] {{nowrap|(''P'', ''L'', ''I'')}} consisting of a set ''P'' of points, a set ''L'' of lines, and an [[incidence relation]] ''I'' stating which points lie on which lines. Obtaining a ''finite'' projective space requires one more axiom: * The set of points ''P'' is a finite set. In any finite projective space, each line contains the same number of points and the ''order'' of the space is defined as one less than this common number. A subspace of the projective space is a subset ''X'', such that any line containing two points of ''X'' is a subset of ''X'' (that is, completely contained in ''X''). The full space and the empty space are always subspaces. The ''geometric dimension'' of the space is said to be ''n'' if that is the largest number for which there is a strictly ascending chain of subspaces of this form: : <math> \varnothing = X_{-1} \subset X_{0}\subset \cdots \subset X_{n} = P .</math>
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