Projective space

Abstract projective space

Let S be a set together with some distinguished subsets. To each distinguished subset an integer −1 ≤𝑑 ≤𝑛 is associated. S is called an abstract projective space of dimension 𝑛, and the subsets are called subspaces of dimension 𝑑, if the following axioms are satisfied¹ geo

1. For every −1 ≤𝑑 ≤𝑛 there is at least one subspace of dimension 𝑑, moreover ^P1
   • ∅ is the unique subspace of dimension −1;
   • S is the unique subspace of dimension 𝑛; and
   • subspaces of dimension 0 are singletons.
2. If a subspace of dimension 𝑟 is contained in a subspace of dimension 𝑠, then 𝑟 ≤𝑠, and 𝑟 =𝑠 iff the subspaces coïncide.
3. The intersection of subspaces is a subspace
4. If the intersection of a subspace of dimension 𝑟 and a subspace of dimension 𝑠 is a subspace of dimension 𝑡, and the intersection of all subspaces containing both of the subspaces is a subspace of dimension 𝑢, then 𝑟 +𝑠 =𝑡 +𝑢.
5. Each subspace of dimension 1 contains 𝑞 +1 ≥3 elements.

Subspaces of dimension 0 are called points; 1 are called lines; 2 are called planes; and 𝑛 −1 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

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Footnotes

1. 2020. Finite geometries, p. 75 ↩