Editing Projective space (section)

== Synthetic geometry ==
In [[synthetic geometry]], a '''projective space''' {{math|''S''}} can be defined axiomatically as a set {{math|''P''}} (the set of points), together with a set {{math|''L''}} of subsets of {{math|''P''}} (the set of lines), satisfying these axioms:<ref>{{harvnb|Beutelspacher|Rosenbaum|1998|pp=6–7}}</ref>
* Each two distinct points {{math|''p''}} and {{math|''q''}} are in exactly one line.
* [[Oswald Veblen|Veblen]]'s axiom:{{efn|also referred to as the ''Veblen–Young axiom'' and mistakenly as the [[Pasch's axiom|axiom of Pasch]] {{harv|Beutelspacher|Rosenbaum|1998|pp=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.}} If {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} are distinct points and the lines through {{math|''ab''}} and {{math|''cd''}} meet, then so do the lines through {{math|''ac''}} and {{math|''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]] {{math|(''P'', ''L'', ''I'')}} consisting of a set {{math|''P''}} of points, a set {{math|''L''}} of lines, and an [[incidence relation]] {{math|''I''}} that states which points lie on which lines.

The structures defined by these axioms are more general than those obtained from the vector space construction given above. If the (projective) dimension is at least three then, by the [[Veblen–Young theorem]], there is no difference. However, for dimension two, there are examples that satisfy these axioms that can not be constructed from vector spaces (or even modules over division rings). These examples do not satisfy the [[theorem of Desargues]] and are known as [[non-Desarguesian plane]]s. In dimension one, any set with at least three elements satisfies the axioms, so it is usual to assume additional structure for projective lines defined axiomatically.<ref>{{harvnb|Baer|2005|p=71}}</ref>

It is possible to avoid the troublesome cases in low dimensions by adding or modifying axioms that define a projective space. {{harvtxt|Coxeter|1969|p=231}} gives such an extension due to Bachmann.<ref>{{citation |first=F. |last=Bachmann |title=Aufbau der Geometrie aus dem Spiegelsbegriff |series=Grundlehren der mathematischen Wissenschaftern, 96 |publisher=Springer |place=Berlin |year=1959 |pages=76–77 }}</ref> To ensure that the dimension is at least two, replace the three point per line axiom above by:
* There exist four points, no three of which are collinear.
To avoid the non-Desarguesian planes, include [[Pappus's hexagon theorem|Pappus's theorem]] as an axiom;{{efn|As Pappus's theorem implies Desargues's theorem this eliminates the non-Desarguesian planes and also implies that the space is defined over a field (and not a division ring).}}
* If the six vertices of a hexagon lie alternately on two lines, the three points of intersection of pairs of opposite sides are collinear.
And, to ensure that the vector space is defined over a field that does not have even [[Characteristic (field)|characteristic]] include ''Fano's axiom'';{{efn|This restriction allows the real and complex fields to be used (zero characteristic) but removes the [[Fano plane]] and other planes that exhibit atypical behavior.}}
* The three diagonal points of a [[complete quadrangle]] are never collinear.

{{anchor|Projective subspace}}A '''subspace''' of the projective space is a subset {{math|''X''}}, such that any line containing two points of {{math|''X''}} is a subset of {{math|''X''}} (that is, completely contained in {{math|''X''}}).  The full space and the empty space are always subspaces.

The geometric dimension of the space is said to be {{math|''n''}} if that is the largest number for which there is a strictly ascending chain of subspaces of this form:
<math display="block">\varnothing = X_{-1}\subset X_{0}\subset \cdots X_{n}=P.</math>

A subspace {{math|''X''<sub>''i''</sub>}} in such a chain is said to have (geometric) dimension {{math|''i''}}. Subspaces of dimension 0 are called ''points'', those of dimension 1 are called ''lines'' and so on. If the full space has dimension {{math|''n''}} then any subspace of dimension {{nowrap|''n'' − 1}} is called a [[hyperplane]].

Projective spaces admit an equivalent formulation in terms of [[lattice (order)|lattice]] theory. There is a bijective correspondence between projective spaces and geomodular lattices, namely, [[subdirectly irreducible]], [[Compact element|compactly generated]], [[complemented lattice|complemented]], [[modular lattice]]s.<ref>Peter Crawley and [[Robert P. Dilworth]], 1973. ''Algebraic Theory of Lattices''. Prentice-Hall. {{isbn|978-0-13-022269-5}}, p. 109.</ref>

=== Classification ===
* Dimension 0 (no lines): The space is a single point.
* Dimension 1 (exactly one line): All points lie on the unique line.
* Dimension 2: There are at least 2 lines, and any two lines meet. A projective space for {{math|1=''n'' = 2}} is equivalent to a [[projective plane]].  These are much harder to classify, as not all of them are isomorphic with a {{math|PG(''d'', ''K'')}}. The [[Desarguesian plane]]s (those that are isomorphic with a {{math|PG(2, ''K''))}} satisfy [[Desargues's theorem]] and are projective planes over division rings, but there are many [[non-Desarguesian plane]]s.
* Dimension at least 3: Two non-intersecting lines exist. {{harvtxt|Veblen|Young|1965}} proved the [[Veblen–Young theorem]], to the effect that every projective space of dimension {{math|''n'' ≥ 3}} is isomorphic with a {{math|PG(''n'', ''K'')}}, the {{math|''n''}}-dimensional projective space over some [[division ring]]&nbsp;{{math|''K''}}.

=== Finite projective spaces and planes ===
{{details|topic=finite projective planes|Projective plane#Finite projective planes}} 
[[Image:Fano plane.svg|thumb|right|The [[Fano plane]] ]]

A ''finite projective space'' is a projective space where {{math|''P''}} is a finite set of points. 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. For finite projective spaces of dimension at least three, [[Wedderburn's little theorem|Wedderburn's theorem]] implies that the division ring over which the projective space is defined must be a [[finite field]], {{math|GF(''q'')}}, whose order (that is, number of elements) is {{math|''q''}} (a prime power). A finite projective space defined over such a finite field has {{math|''q'' + 1}} points on a line, so the two concepts of order coincide. Notationally, {{math|PG(''n'', GF(''q''))}} is usually written as {{math|PG(''n'', ''q'')}}.

All finite fields of the same order are isomorphic, so, up to isomorphism, there is only one finite projective space for each dimension greater than or equal to three, over a given finite field. However, in dimension two there are non-Desarguesian planes. Up to isomorphism there are
{{block indent | em = 1.5 | text = 1, 1, 1, 1, 0, 1, 1, 4, 0, ... {{OEIS|id=A001231}}}}
finite projective planes of orders 2, 3, 4, ..., 10, respectively. The numbers beyond this are very difficult to calculate and are not determined except for some zero values due to the [[Bruck–Ryser–Chowla theorem|Bruck–Ryser theorem]].

The smallest projective plane is the [[Fano plane]], {{math|PG(2, 2)}} with 7 points and 7 lines. The smallest 3-dimensional projective space is [[PG(3,2)|{{math|PG(3, 2)}}]], with 15 points, 35 lines and 15 planes.