Abstract projective space

Let $S$ be a set together with some distinguished subsets. To each distinguished subset an integer $- 1 \leq 𝑑 \leq 𝑛$ 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 #m/def/geo

For every $-1 \leq d \leq n$ there is at least one subspace of dimension $d$ , moreover
- $\emptyset$ is the unique subspace of dimension $- 1$ ;
- $S$ is the unique subspace of dimension $𝑛$ ; and
- subspaces of dimension $0$ are singletons.
If a subspace of dimension $𝑟$ is contained in a subspace of dimension $𝑠$ , then $𝑟 \leq 𝑠$ , and $𝑟 = 𝑠$ iff the subspaces coïncide.
The intersection of subspaces is a subspace
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 $𝑟 + 𝑠 = 𝑡 + 𝑢$ .
Each subspace of dimension 1 contains $𝑞 + 1 \geq 3$ 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

#state/tidy | #lang/en | #SemBr

2020. Finite geometries, p. 75 ↩

Abstract projective space

Further terminology

Footnotes