Domain in the complex plane

Definition

Let $D$ be a subset of the complex plane $\mathbb{C}$. The set $D$ is open if for each $z_0 \in D$ there is a real number $r > 0$ such that each $z \in \mathbb{C}$ with $|z - z_0| < r$ is in $D$. Here $|z - z_0|$ is the modulus of $z - z_0$. The set $D$ is connected if there are no two disjoint nonempty open sets $A$ and $B$ with $D = A \cup B$. A domain in the complex plane is a subset $D \subseteq \mathbb{C}$ that is nonempty, open and connected.

Remarks

An open set $D \subseteq \mathbb{C}$ is connected if and only if each two points of $D$ are the end points of a polygonal path in $D$. Examples of domains are $\mathbb{C}$, each open disc, the upper half-plane $\{z : \operatorname{Im} z > 0\}$ and $\mathbb{C} \setminus \{0\}$. The union of two disjoint open discs is open but not connected, so it is not a domain. Some authors use the word region for a domain. Other authors use region for a set between a domain and its closure.

Explanations

L1 human

A domain is a set in the plane that has no boundary points in it and that is in one piece. Open means that you can move a little in each direction from each point and stay in the set. Connected means that you can walk from each point of the set to each other point without leaving the set.