Projective space

Abstract projective space

Let be a set together with some distinguished subsets. To each distinguished subset an integer is associated. is called an abstract projective space of dimension , and the subsets are called subspaces of dimension , if the following axioms are satisfied¹ geo

1. For every there is at least one subspace of dimension , moreover ^P1
   • is the unique subspace of dimension ;
   • is the unique subspace of dimension ; and
   • subspaces of dimension are singletons.
2. If a subspace of dimension is contained in a subspace of dimension , then , and iff the subspaces coïncide.
3. The intersection of subspaces is a subspace
4. If the intersection of a subspace of dimension and a subspace of dimension is a subspace of dimension , and the intersection of all subspaces containing both of the subspaces is a subspace of dimension , then .
5. Each subspace of dimension 1 contains elements.

Subspaces of dimension 0 are called points; 1 are called lines; 2 are called planes; and are called hyperplanes. This generalizes the Abstract projective plane. See Finite projective space.

Further terminology

• An isomorphism of projective spaces is a Collineätion.
• A duality map of a projective space is a Projective correlation

---

tidy | en | sembr

Footnotes

1. 2020. Finite geometries, p. 75 ↩