In geometry, we have (kind of) introduced the projective space. Sadly, I have problems understanding some connections and I hope somebody here might help me out, as wikipedia's entry and my professor's notes were of no avail.
Let $K$ be a field. We call $\mathbb{A}^n(K) := K^n$ the affine space over $K$ with dimension $n \in \mathbb{N}_0.$
Question 1: What properties does an affine space have? This is the first time our professor used this notion and I don't see how this "definition" really defines an affine space. German wikipedia tells me it is just some space where we have points, lines and some axioms (only naming the parallel axiom). Which other axioms hold in an affine space?
We say that $x,y \in K^{n+1} \setminus \{0\}$ are equivalent, if $\exists t \in K \setminus \{0\}: y = tx$. This is an equivalence relation. Let $\mathbb{P}^n(K)$ denote the set of equivalence classes. So $\mathbb{P}^n(K)$ is the set of all "terms" $(x_0: \dots : x_n)$ where $x_0, ..., x_n \in K$ and not all $x_0, ..., x_n$ are zero. We see that the map
$\mathbb{A}^n(K) \to \{(x_0: \dots:x_n) \in \mathbb{P}^n(K) | \; x_0 \neq 0\}, (x_1, \dots, x_n) \mapsto (1:x_1:\dots:x_n)$
and the complement
$\mathbb{P}^{n-1}(K) \to \{(0:x_1:\dots:x_n) \in \mathbb{P}^n(K)\}, (x_1 : \dots : x_n) \mapsto (0:x_1:\dots:x_n)$
are bijective.
Question 2: I can see why these two functions are bijective. But what exactly do we need them for? What is their meaning? And why is the second map called "the complement"?
We have $\mathbb{P}^1(K) = \mathbb{A}^1(K) \sqcup \mathbb{P}^0(K)$, where $\mathbb{P}^1(K)$ is a projective line and $\mathbb{P}^0(K) = \{\infty\}$.
Question 3: How does one obtain this equality? I mean $\mathbb{P}^1(K)$ is a set containing equivalence classes whereas $\mathbb{A}^1(K)$ is a set only containing 1-dimensional points. Also, why is $\mathbb{P}^0(K)=\{\infty\}$?
Thank you very much in advance for any answers.