I want to prove following (Big Picard Theorem forms):
Theorem.The followings are equivalent:
a) If $f \in H(\mathbb{D}\setminus\{0\})$ and $f(\mathbb{D}\setminus\{0\}) \subset \mathbb{C} \setminus \{0, 1\}$, then $f$ has a pole or an removable singularity at $0$.
b) Let $\Omega \subset \mathbb{C}$ is a open subset, $f : \Omega \to \mathbb{C}$ is holomorphic and $z_0 \in \mathbb{C}$. If $f$ has an essential singularity at $z_0$, then, with at most one exception, $f$ attains every complex value infinitely many times;
c) Let $f : \mathbb{C} \to \mathbb{C}$ a entire function which is not polynomial. Then, with at most one exception, $f$ attains every complex value infinitely many times;
I have proved that a)$\implies$b)$\implies$c) and that b)$\implies$a) but I don't know how to start proving that c) implies a) or b). And another thing: Is mathematically correct to say that those points are equivalent? I only want a hint or something that can make me understand how to start. Thanks! I see that in a) and b) I have a open set $\Omega$, but in c) I have the full complex plane, and I don't know how to reduce from $\mathbb{C}$ to $\Omega$
Edit: Here is a photo from the textbook (Page $358$):