📂Complex Anaylsis

Topology of Complex Spaces

Overview

We introduce definitions for dealing with the set of complex numbers $\mathbb{C}$ as a topological space. Although it is referred to as a topological space, most of the definitions are specializations of the definitions in a metric space for complex sets. If you have studied introductory analysis diligently, you will be able to understand these without much difficulty.

Definitions ¹

Let’s assume $\alpha \in \mathbb{C}$, $\delta > 0$ and $S \subset \mathbb{C}$.

Open and Closed Sets

1. The following set is called the Open Neighborhood or Open Ball of $\alpha$. $$B \left( \alpha ; \delta \right) := \left\{ z \in \mathbb{C} : \left| z - \alpha \right| < \delta \right\}$$ When an asterisk $\ast$ is superscripted, it means that the center $\alpha$ is excluded. For example, $B^{\ast} \left( \alpha ; \delta \right)$ is defined as follows and is called a Punctured Ball. $$B^{\ast} \left( \alpha ; \delta \right) := \left\{ z \in \mathbb{C} : 0 < \left| z - \alpha \right| < \delta \right\}$$
2. If any open ball of $\alpha$ is included in $S$, then $\alpha$ is called an Interior Point of $S$. $$\exist \delta : B \left( \alpha , \delta \right) \subset S$$
3. If every punctured open ball of $\alpha$ is not disjoint with $S$, then $\alpha$ is called a Limit Point of $S$. $$\forall \delta : B^{\ast} \left( \alpha , \delta \right) \cap S \ne \emptyset$$
4. If every point of $S$ is an interior point of $S$, then $S$ is said to be Open; and if $S$ contains all its limit points, it is said to be Closed.

Bounded and Compact

5. If for every element $z \in S$ of $S \subset \mathbb{C}$ there exists a positive number $M > 0$ that satisfies $\left| z \right| \le M$, then $S$ is said to be Bounded.
6. If it is closed and bounded, it is called Compact.

Complex Domain

7. If every two points of $S \subset \mathbb{C}$ can be connected by segments forming a path, then $S$ is called a (Polygonally) Connected set.
8. A non-empty, open connected set $\mathscr{R} \subset \mathbb{C}$ is called a Region, and particularly in the context of complex space, it is emphasized as a Complex Region.

Further Definitions

This section summarized parts universally essential in mathematics, not just in complex analysis. Of course, the following definitions and notations are also necessary when needed.

9. The following set is called the Closed Neighborhood or Closed Ball of $\alpha$. $$B \left[ \alpha ; \delta \right] := \left\{ z \in \mathbb{C} : \left| z - \alpha \right| \le \delta \right\}$$
10. If every open neighborhood of $\alpha$ contains a point of $S$ and $S^{c}$, then $\alpha$ is called a Boundary Point. If $\alpha$ is neither an interior point nor a boundary point, it is called an Exterior Point.
11. The set of all limit points of $S$ is called the Closure of $S$, and it is represented as $\overline{S}$.
12. If $\mathbb{C} \setminus S$ is a connected set, then the connected set $S$ is called Simply Connected.

See Also

The set of complex numbers $\mathbb{C}$ not only follows the axioms of a field but also is a $\mathbb{C}$-vector space with the modulus of complex numbers $\left| \cdot \right|$, making it a normed space as well as a metric space. Therefore, if you are already familiar with metric spaces, there is nothing new to learn specifically as a complex space.

• Balls and Open and Closed Sets in Metric Spaces
• Neighborhoods, Accumulation Points, Openness, and Closure in Metric Spaces
• Interior, Closure, and Boundary in Metric Spaces
• Compactness in Metric Spaces
• Heine-Borel Theorem Proof: Originally, defining a compact requires much more complicated discussions but in complex analysis, it’s acceptable to define compactness simply as equivalence if it is bounded and a closed set.
• Path Connectivity in Topology: Actually, the connectivity introduced in the definition is closer to path connectivity, because if it’s path-connected, it is connected, and to understand the general definition of connectivity in topology, a solid topological mindset is required, hence, a geometric intuition of ‘being connected by segments’ is borrowed instead.
• Connected Sets in Metric Spaces