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 :

- Each two distinct points
*p*and*q*are in exactly one line. - Veblen's axiom: 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 (*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.

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:

Read more about Projective Space: Morphisms, Dual Projective Space, Generalizations

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### Famous quotes containing the word space:

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