Consider the function $f: \mathbb{C} \rightarrow \mathbb{C}$ given by $f(z)=e^{z}$. Which of the following is false?

1. $f(\{z \in \mathbb{C}:|z|<1\})$ is not an open set.

2. $f(\{z \in \mathbb{C}:|z| \leq 1\})$ is not an open set.

3. $f(\{z \in \mathbb{C}:|z|=1\})$ is a closed set.

4. $f(\{z \in \mathbb{C}:|z|>1\})$ is an unbounded open set.