I am looking at various proofs of the Little and Big Picard Theorems. I am interested in the following question:
Without the Uniformization Theorem, can one calculate the Euler characteristic of $\mathbb{C} \backslash \{0,1\}$? In general, how does one calculate the Euler characteristic of non-compact surfaces? Is there a formula?
I ask because, if one knew the Euler characteristic of $\mathbb{C} \backslash \{0,1\}$, then one could apply the Uniformization Theorem to conclude that $\mathbb{C}\backslash \{0,1\}$ has the unit disc as its universal cover. Lifting an entire function $f: \mathbb{C} \to \mathbb{C}\backslash \{0,1\}$ to $\tilde{f}: \mathbb{C} \to U$ we obtain an "easy" proof of the Little Picard Theorem. Of course, this requires doing the work of proving the Uniformization Theorem.