The set $S$ of complex numbers $z$ satisfying $|z| < 1$ or $| z − 3i | < 1$ is a domain. Is this statement true or false?

Question

The set $S$ of complex numbers z satisfying $|z| < 1$ or $| z − 3i | < 1$ is a domain.

I found this statement true but the book (Dennis G. Zill and Patrick D. Shanahan - A First Course in Complex Analysis with Applications (2003, Jones and Bartlett Publishers, Inc.) says it is false.

I tried to look at it in this way. Firstly, the $0$ centered open circle is a domain because it is open and connected. Secondly, the $0+3i$ centered ($(0,3)$ on complex plane) circle is also open and connected. I said this statement is true but the book says it is false, why?

Answer 1

The set $S$ is explicitly described as \begin{equation} S = \{ z \in \mathbb{C} : |z| < 1 \text{ or } |z - 3i| < 1\} \end{equation} which is open, but not connected.

Answer 2

It is true that the two disks described are open connected sets, but they are 1 unit apart so their union is (open but) not connected.