---
tag: H30E38799N
kind: definition
name: Projective space
msc: [14N05, 51N15]
defines: ZT9A6JK8QF
uses: [04C5ZBG1WT, 000AYPA9AJ, 0G8VEZ3WTK, 00YC1D652K, 1DDYB6VVSR]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_space
---

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $n \ge 0$ be an integer.
On $k^{n+1} \setminus \set{0}$, let $x \sim y$ when $y = \lambda x$ for some $\lambda \in k \setminus \set{0}$; this is an [[equivalence relation|0G8VEZ3WTK]].
The *projective $n$-space* over $k$ is the set of equivalence classes
$$\mathbb{P}^n = \mathbb{P}^n(k) = \left(k^{n+1} \setminus \set{0}\right) / \sim.$$
The class of $(x_0, \dots, x_n)$ is written $[x_0 : \dots : x_n]$, and the $x_i$ are its homogeneous coordinates.

## Remarks

Equivalently, $\mathbb{P}^n$ is the set of one-dimensional [[linear subspaces|00YC1D652K]] of the [[vector space|000AYPA9AJ]] $k^{n+1}$.
More generally, $\mathbb{P}(V)$ denotes the set of one-dimensional subspaces of a vector space $V$.

Example: $[1 : 2 : 3] = [2 : 4 : 6]$ in $\mathbb{P}^2(\Q)$, because $(2, 4, 6) = 2 \cdot (1, 2, 3)$.
Over $\R$ with its usual topology, $\mathbb{P}^n(\R)$ is the [[real projective space|1DDYB6VVSR]].

Non-example: $(0 : 0 : 0)$ is not a point of $\mathbb{P}^2$, since the zero vector is excluded.

For each $i$, the set $U_i = \set{x_i \ne 0}$ is in bijection with $k^n$ through $[x_0 : \dots : x_n] \mapsto (x_0/x_i, \dots, \widehat{x_i/x_i}, \dots, x_n/x_i)$, and the $U_i$ cover $\mathbb{P}^n$.
The complement of $U_0$ is $\set{x_0 = 0}$, a copy of $\mathbb{P}^{n-1}$, the hyperplane at infinity.