Euclid's parallel postulate asserts that for every a pair of points and lines $(p,L)$ in [[Affine Space]], there exists a unique line $L'$ parallel to $L$ passing through $p$ (if $p$ lies on $L$ then $L'=L$). Hyperbolic space is an example a metric space that violates this postulate: there are infinitely many points parallel to $L$ passing through $p$. Projective space is another space that violates the parallel postulate in that any to lines intersect, that is, no two lines are parallel. Projective space is central in algebraic geometry. Projective varieties have the Zariski topology. # Definitions > [!NOTE] Definition > The projective $n$-space over $K$ is the set of orbits in $K^{n+1}\setminus \{ 0 \}$ under the [[Group action|action]] of the [[Unit Group of Ring|unit group]] $K^{\times}$ by scalar multiplication. We verify that scalar multiplication indeed defines group action: for $v\in K^{n+1}\setminus\{ 0 \}$, $1\cdot v = v$ and for all $\lambda_{1},\lambda_{2}\in K^{\times}$, $(\lambda_{1}\lambda_{2})v=\lambda_{1}(\lambda_{2}v)$ (these follow from the last two vector space axioms). **Remarks**: - i.e. two 'position vectors' are equivalent if there are scalar multiples of each other: bijection to set of lines through the origin - the elements of $\mathbb{P}^n$ (equivalence classes/ affine lines through the origin) are called points. - the list $(x_{1},x_{2},\dots,x_{n+1})$ is a (well-defined) function on $K^{n+1}$ not $K\mathbb{P}^n$; the things that are functions are ideals of homogeneous polynomials of the same degree - projective planes are projective $2$-spaces > [!Example]- $1$-dimensional $\mathbb{R}$ projective space > We have $\begin{align} \mathbb{P}^1 = \mathbb{P}(\mathbb{R}^2) &= \{ (x:1) \mid x\in \mathbb{R} \} \cup \{ (1:0) \}\\&= \{ (\cos\theta: \sin \theta ) \mid \theta\in [0,\pi) \} \end{align}$ > That is, the projective space is the real line together with the 'point at infinity', which of course doesn't ordinarily admit any viewpoint in space. completion > [!NOTE] Definition (Axiomatic/synthetic definition) > An axiomatic projective plane is a tuple $(P,L,I)$ where: $P$ is seen as a set of points; $L$, a set of lines; and $I \subset P\times L$, the incidence relation; satisfying the following axioms: > > 1. Every line contains at least three points: for all $\ell \in L$, there exist distinct $x_{1},x_{2},x_{3}\in P$ such that $(x_{i}, \ell)\in I$. > 2. Every point is contained in at least three distinct lines > 3. Any two points span a unique line. > 4. Any two distinct lines intersect a unique point. # Properties Existence of bijection between lines through origin in $\mathbb{R}^{n+1}$ and elements of $\mathbb{P}^n=\mathbb{P}(\mathbb{R}^{n+1})$.