Open Sets in the Complex Plane

Open Sets in the Complex Plane

The complex plane can be regarded as a topological space that is homeomorphic to $\mathbb{R}^2$ with the usual Euclidean topology, or a metric space $(\mathbb{C}, d)$ where $d : \mathbb{C} \times \mathbb{C} \to [0, \infty)$ is defined for all $z, w \in \mathbb{C}$ by $\mid z - w \mid$, i.e., the modulus of the difference $z - w$. Nevertheless, we will review these concepts.

Definition: Let $z \in \mathbb{C}$ and $r > 0$. The Open Disk of radius $r$ centered at $z$ is defined to be the set $D(z, r) = \{ y \in \mathbb{C} : \mid z - y \mid < r \}$.
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With the notion of open disks we can define open sets in $\mathbb{C}$.

Definition: Let $A \subseteq \mathbb{C}$. $A$ is said to be Open in $\mathbb{C}$ if for every $z \in A$ there exists an $r > 0$ such that $D(z, r) \subseteq A$.

Equivalently, $A$ is said to be open if $A = \mathrm{int} (A)$.

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Trivially, the empty set $\emptyset$ and whole set $\mathbb{C}$ are open sets.

With these two notions, it can be shown that $\mathbb{C}$ is a topological space.

Proposition 1: The open sets of $\mathbb{C}$ satisfy the following properties:
a) $\emptyset$ and $\mathbb{C}$ are open in $\mathbb{C}$.
b) If $\{ U_i : i \in I \}$ is an arbitrary collection of open sets in $\mathbb{C}$ then $\displaystyle{\bigcup_{i \in I} U_i}$ is open in $\mathbb{C}$.
c) If $\{ U_1, U_2, ..., U_n \}$ is a finite collection of open sets in $\mathbb{C}$ then $\displaystyle{\bigcap_{i=1}^{n} U_i}$ is open in $\mathbb{C}$.

Property (b) can be summarized to say that a union of an arbitrary collection of open sets in $\mathbb{C}$ is also open in $\mathbb{C}$, while property (c) can be summarized to say that an intersection of a finite collection of open sets in $\mathbb{C}$ is also open in $\mathbb{C}$.

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