Topology of the Complex Plane

The complex plane $\mathbb{C}$ inherits a natural metric topology from the Euclidean distance $|z - w|$ . Understanding open sets, domains, and connectedness is essential for stating theorems precisely in complex analysis.

Metric and Open Sets

Definition1.8Metric

The distance between $z, w \in \mathbb{C}$ is

$d(z, w) = |z - w| = \sqrt{(x - u)^2 + (y - v)^2}$

where $z = x + iy$ and $w = u + iv$ . This makes $\mathbb{C}$ a metric space.

Definition1.9Open disk

The open disk of radius $r > 0$ centered at $z_0 \in \mathbb{C}$ is

$D(z_0, r) = \{z \in \mathbb{C} \mid |z - z_0| < r\}.$

The closed disk is $\overline{D}(z_0, r) = \{z \mid |z - z_0| \leq r\}$ .

Definition1.10Open set

A set $U \subseteq \mathbb{C}$ is open if for every $z_0 \in U$ , there exists $r > 0$ such that $D(z_0, r) \subseteq U$ . Equivalently, $U$ is a union of open disks.

ExampleExamples of open sets

Any open disk $D(z_0, r)$ is open.
The right half-plane $\{\text{Re}(z) > 0\}$ is open.
The punctured plane $\mathbb{C} \setminus \{0\}$ is open.
The unit circle $\{|z| = 1\}$ is not open (it contains no open disk).
The closed disk $\{|z| \leq 1\}$ is not open.

RemarkClosed sets

A set $F \subseteq \mathbb{C}$ is closed if its complement $\mathbb{C} \setminus F$ is open. Equivalently, $F$ contains all its limit points.

Domains and Regions

Definition1.11Domain

A domain is a nonempty open connected subset of $\mathbb{C}$ . That is, $D \subseteq \mathbb{C}$ is a domain if:

$D$ is open.
$D$ is connected (cannot be written as a union of two disjoint nonempty open sets).

ExampleExamples of domains

The unit disk $D(0, 1) = \{z \mid |z| < 1\}$ is a domain.
The punctured disk $D(0, 1) \setminus \{0\}$ is a domain.
The right half-plane $\{\text{Re}(z) > 0\}$ is a domain.
The set $\{|z| < 1\} \cup \{|z - 3| < 1\}$ (two disjoint disks) is not a domain (not connected).

Definition1.12Region

A region is a domain together with some, none, or all of its boundary points. More generally, a region is the closure of a domain.

Connectedness and Simply Connected

Definition1.13Connected set

A set $E \subseteq \mathbb{C}$ is connected if it cannot be written as $E = A \cup B$ where $A$ and $B$ are nonempty, disjoint, and relatively open in $E$ .

RemarkPath-connected vs. connected

In $\mathbb{C}$ (and more generally in $\mathbb{R}^n$ ), connected and path-connected coincide for open sets. Thus an open set $U \subseteq \mathbb{C}$ is connected if and only if any two points in $U$ can be joined by a continuous curve lying entirely in $U$ .

Definition1.14Simply connected

A domain $D \subseteq \mathbb{C}$ is simply connected if every closed curve in $D$ can be continuously shrunk to a point within $D$ . Equivalently, $D$ has no "holes."

ExampleSimply connected vs. multiply connected

The unit disk $\{|z| < 1\}$ is simply connected.
The complex plane $\mathbb{C}$ is simply connected.
The right half-plane $\{\text{Re}(z) > 0\}$ is simply connected.
The annulus $\{1 < |z| < 2\}$ is not simply connected (the unit circle cannot be shrunk to a point within the annulus).
The punctured disk $\{0 < |z| < 1\}$ is not simply connected (a small circle around the origin cannot be shrunk).
The punctured plane $\mathbb{C} \setminus \{0\}$ is not simply connected.

Boundary and Compact Sets

Definition1.15Boundary

The boundary of a set $E \subseteq \mathbb{C}$ is

$\partial E = \overline{E} \cap \overline{\mathbb{C} \setminus E}$

where $\overline{E}$ denotes the closure of $E$ (the smallest closed set containing $E$ ).

ExampleExamples of boundaries

The boundary of the open disk $D(0, 1)$ is the unit circle $\{|z| = 1\}$ .
The boundary of the closed disk $\{|z| \leq 1\}$ is also the unit circle.
The boundary of the right half-plane $\{\text{Re}(z) > 0\}$ is the imaginary axis $\{\text{Re}(z) = 0\}$ .
The boundary of the punctured disk $\{0 < |z| < 1\}$ is $\{|z| = 1\} \cup \{0\}$ (both the circle and the origin).

Definition1.16Compact set

A set $K \subseteq \mathbb{C}$ is compact if every open cover of $K$ has a finite subcover. By the Heine-Borel theorem, $K$ is compact if and only if it is closed and bounded.

ExampleExamples of compact sets

The closed disk $\{|z| \leq 1\}$ is compact.
The unit circle $\{|z| = 1\}$ is compact.
The open disk $\{|z| < 1\}$ is not compact (bounded but not closed).
The closed half-plane $\{\text{Re}(z) \geq 0\}$ is not compact (closed but not bounded).
The set $\{1/n \mid n \in \mathbb{N}\} \cup \{0\}$ is compact.

Summary

RemarkKey topological notions

Open sets and domains: domains are the natural setting for complex analysis.
Connected vs. simply connected: connectedness means "in one piece," while simple connectedness means "no holes."
Compact sets: closed and bounded. Many theorems require compactness (e.g., maximum modulus).
Boundary behavior: understanding how functions behave near boundaries is central (e.g., Cauchy integral formula, Schwarz reflection principle).

These topological concepts underpin all major theorems in complex analysis, from Cauchy's theorem to Riemann mapping theorem.