1.3Sets in the Complex Plane

Let $z_0 \in \mathbb{C}$ and $r > 0$.

1. The open disc of radius $r$ centered at $z_0$ is:

   $$D_r(z_0) = \{ z \in \mathbb{C}: |z - z_0| < r \}$$

2. The closed disc of radius $r$ centered at $z_0$ is:

   $$\overline{D}_r(z_0) = \{ z \in \mathbb{C}: |z - z_0| \leq r \}$$

3. The boundary of either diskis the circle $C_r(z_0) = \{ z \in \mathbb{C}: |z - z_0| = r \}$.

4. The unit disc is denoted by $\mathbb{D} = \{ z \in \mathbb{C}: |z| < 1 \}$.

Let $\Omega \subset \mathbb{C}$ be a set.

1. A point $z_0 \in \Omega$ is an interior point of $\Omega$ if there exists $r > 0$ such that $D_r(z_0) \subset \Omega$.

2. The interior of $\Omega$, denoted $\Omega^\circ$, is the set of all its interior points.

3. $\Omega$ is open if every point in $\Omega$ is an interior point (i.e., $\Omega = \Omega^\circ$).

4. $\Omega$ is closed if its complement $\Omega^c = \mathbb{C} \setminus \Omega$ is open.

5. $z \in \mathbb{C}$ is a limit point of $\Omega$ if there is a sequence $\{z_n\} \subset \Omega$ with $z_n \neq z$ such that $z_n \to z$. A set is closed iff it contains all its limit points.

6. The closure of $\Omega$ is $\overline{\Omega} = \Omega \cup \{ \text{limit points} \}$.

7. The boundary of $\Omega$ is $\partial \Omega = \overline{\Omega} \setminus \Omega^\circ$.

8. $\Omega$ is bounded if there exists $M > 0$ such that $|z| < M$ for all $z \in \Omega$.

9. If $\Omega$ is bounded, its diameter is:

   $$\text{diam}(\Omega) = \sup_{z, w \in \Omega} |z-w|$$

A set $\Omega \subset \mathbb{C}$ is compact if it is closed and bounded.

Let $\Omega \subset \mathbb{C}$ be a subset. The following are equivalent:

1. $\Omega$ is compact (closed and bounded).

2. Every sequence $\{z_n\} \subset \Omega$ has a subsequence that converges to a point in $\Omega$ (sequential compactness).

3. Every open covering of $\Omega$ admits a finite subcovering.

If $\Omega_1 \supset \Omega_2 \supset \cdots \supset \Omega_n \supset \cdots$ is a nested sequence of non-empty compact sets in $\mathbb{C}$ with $\text{diam}(\Omega_n) \to 0$ as $n \to \infty$, then there exists a unique point $w \in \mathbb{C}$ such that $w \in \Omega_n$ for all $n$.

An open set $\Omega \subset \mathbb{C}$ is connected if it is not possible to find two disjoint, non-empty open sets $\Omega_1, \Omega_2$ such that $\Omega = \Omega_1 \cup \Omega_2$.

1. A connected open set in $\mathbb{C}$ is called a region.

2. For open sets in $\mathbb{C}$, connectedness is equivalent to path-connectedness: any two points in $\Omega$ can be joined by a continuous curve entirely contained in $\Omega$.