Projective space

Remarks

Equivalently, $\mathbb{P}^n$ is the set of one-dimensional linear subspaces of the vector space $k^{n+1}$. 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.

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.