📂Complex Anaylsis

Topology of Complex Spaces

Overview

Introduce definitions for treating the set $\mathbb{C}$ of complex numbers as a topological space. Although called a topological space, most definitions are specializations of the definitions in a metric space to the complex set. If you have studied analysis thoroughly, these should not be hard to accept.

Definition ¹

Assume $\alpha \in \mathbb{C}$, $\delta > 0$, and $S \subset \mathbb{C}$.

Openness and closedness of sets

1. The following set is called an 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 $\ast$ appears as a superscript it means 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 some open ball of $\alpha$ is contained 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 about $\alpha$ intersects $S$ (i.e., is not disjoint from $S$), 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 of its limit points then it is said to be closed. The collection of all open sets, i.e. $\mathcal{T}$, is called the usual (standard) topology of the complex plane. $$ \mathcal{T} := \left\{ U \subset \mathbb{C} : \forall z \in U, \exists \delta > 0 \text{ such that } B(z, \delta) \subset U \right\} $$

Boundedness and compactness

5. We call $S$ bounded if there exists a positive number $M > 0$ such that $\left| z \right| \le M$ holds for every element $z \in S$ of $S \subset \mathbb{C}$.
6. A set that is closed and bounded is called compact.

Complex regions

7. If any two points of $S \subset \mathbb{C}$ can be connected by a path composed of line segments, then $S$ is called (polygonally) connected.
8. A nonempty, open, connected set $\mathscr{R} \subset \mathbb{C}$ is called a region; to emphasize that it is in the complex plane we call it a complex region.

Definitions

The previous paragraph summarized only those parts that are universally essential in mathematics rather than specific to complex analysis. Naturally, the following definitions and notations are also needed at times.

9. The following set is called a 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 points of both $S$ and $S^{c}$, then $\alpha$ is a boundary point of $S$. If $\alpha$ is neither an interior point nor a boundary point, it is an exterior point.
11. The set of all limit points of $S$ is called the closure of $S$ and is denoted $\overline{S}$.
12. If $\mathbb{C} \setminus S$ is a connected set, then the connected set $S$ is called simply connected.

See also

The set $\mathbb{C}$ of complex numbers not only satisfies the field axioms but is also a $\mathbb{C}$-vector space ($\mathbb{C}$-vector space), and with the modulus $\left| \cdot \right|$ it is a normed space and hence a metric space. Therefore, if you are already familiar with metric spaces, there is nothing essentially new to learn specifically about the complex plane.

• Balls and open/closed sets in metric spaces
• Neighborhoods, limit points, openness, closedness in metric spaces
• Interior, closure, boundary in metric spaces
• Compactness in metric spaces
• Proof of the Heine–Borel theorem: To define compactness rigorously one generally needs a more involved discussion, but in complex analysis it is safe to use the characterization that compactness is equivalent to being closed and bounded.
• Path connectedness in topology: The connectivity introduced in the definition above is actually closer to path connectedness; path connectedness implies connectedness, and to understand the general topological definition of connectedness (see Connectedness in topology) requires a fairly solid topological viewpoint, so we borrow the geometric intuition of being connectable by line segments instead.
• Connected sets in metric spaces