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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