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
- 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. $$ B^{\ast} \left( \alpha ; \delta \right) := \left\{ z \in \mathbb{C} : 0 < \left| z - \alpha \right| < \delta \right\} $$
- If any open ball of $\alpha$ is included in $S$, then $\alpha$ is called an Interior Point of $S$. $$ \exist \delta : B \left( \alpha , \delta \right) \subset S $$
- If every punctured open ball of $\alpha$ is not disjoint with $S$, then $\alpha$ is called a Limit Point of $S$. $$ \forall \delta : B^{\ast} \left( \alpha , \delta \right) \cap S \ne \emptyset $$
- 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
- 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.
- If it is closed and bounded, it is called Compact.
Complex Domain
- 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.
- 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.
- The following set is called the Closed Neighborhood or Closed Ball of $\alpha$. $$ B \left[ \alpha ; \delta \right] := \left\{ z \in \mathbb{C} : \left| z - \alpha \right| \le \delta \right\} $$
- 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.
- The set of all limit points of $S$ is called the Closure of $S$, and it is represented as $\overline{S}$.
- 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
Osborne (1999). Complex variables and their applications: p10~12. ↩︎