Consider the two subsets: $X_1=\mathbb P_1(\mathbb C)\setminus\{0,1,\infty, e \}$ and $X_2=\mathbb P_1(\mathbb C)\setminus\{0,1,\infty, \pi \}$. They are two varieties in the sense of the first chapter of Hartshorne (I'd like to avoid the scheme theory here), but I don't understand why they are not isomorphic.
I need the simplest reason that explain the absence of an isomorphism, so a motivation that involves less theory possible.
Thanks in advance.
Suppose $f : X_1 \to X_2$ is an isomorphism. Then it induces a birational map $\bar{f} : \mathbb{P}^1 \to \mathbb{P}^1$, and since $\mathbb{P}^1$ is a smooth projective curve, this birational map must in fact be an automorphism. The automorphisms of $\mathbb{P}^1$ are the Möbius transformations, and these preserve cross-ratios. Clearly, $\bar{f}$ must send $\{ 0, 1, \infty, e \}$ to $\{ 0, 1, \infty, \pi \}$, but the cross-ratios of these quadruples are different – a contradiction.
