Complex Analysis

Domains in Complex Analysis — Open/Closed, Simply/Multiply Connected

A clear, visual guide with static illustrations to build intuition

1) Core definitions

• Open set. \(U \subset \mathbb{C}\) is open if for every \(z \in U\) there exists \(r>0\) such that the open disk \(B(z,r)=\{\,w : |w-z| \lt r\,\} \subset U\).
• Closed set. \(F \subset \mathbb{C}\) is closed if it contains all its limit points (equivalently, its complement is open).
• Domain. A nonempty, open, connected subset of \(\mathbb{C}\).
• Simply connected. A domain \(\Omega\) is simply connected if every closed loop in \(\Omega\) can be continuously contracted to a point within \(\Omega\) (i.e., any loop is homotopic to a constant loop).
• Multiply connected. Not simply connected (intuitively, \(\Omega\) has one or more “holes”).

Planar criterion. For open sets \(\Omega \subset \mathbb{C}\), “simply connected” is equivalent to the complement being connected in the Riemann sphere \(\widehat{\mathbb{C}}\) (the plane plus a point at infinity). The exterior of a disk highlights why the point at infinity matters.

2) Visual vocabulary

Conventions: Inside (region) is a semi-transparent color unique to each figure. The same color is used for the boundary: solid line = boundary included; dashed line = boundary excluded.

Open Disk — Open Simply
Interior only; boundary is not included (dashed).

Closed Disk — Closed Simply
Interior plus boundary; no holes.

Annulus — Open Multiply
A ring-shaped open set; loop around the hole cannot contract.

Punctured Disk — Open Multiply
Missing center forms a “hole”; loops around it cannot contract.

Closed Rectangle — Closed Simply
Includes its boundary; no holes, so simply connected.

Whole Plane ℂ — Open Closed Simply
In the standard topology on ℂ, both open and closed (“clopen”).

3) Open vs. closed — the epsilon-ball test

A point \(z\) is interior to \(U\) if there is some \(r \gt 0\) with \(B(z,r) \subset U\). An open set is one where every point is interior. A point \(z\) is a boundary point of \(U\) if every \(B(z,r)\) meets both \(U\) and its complement. A set is closed if it contains all its limit points (equivalently, if \(\mathbb{C}\setminus F\) is open).

\[ U \text{ open } \iff \forall z\in U \;\exists r\gt 0:\; B(z,r)\subset U, \qquad F \text{ closed } \iff \mathbb{C}\setminus F \text{ is open.} \]

• The open disk excludes its circle boundary; the closed disk includes it.
• The annulus excludes both inner and outer circles (so it’s open).
• Closed rectangle includes its edges; it is closed.

4) Simply vs. multiply connected — loop intuition

Picture a closed loop \(\gamma: [0,1]\to\Omega\) drawn inside a region. If we can shrink \(\gamma\) to a point while staying in \(\Omega\), the loop is null-homotopic. A domain is simply connected if every closed loop is null-homotopic. If some loop “gets stuck” around a hole, the region is multiply connected.

\[ \Omega \text{ simply connected } \iff \pi_1(\Omega) \cong 0 \iff \text{every closed loop contracts within } \Omega. \]

• Open/closed disks and closed rectangles: no holes → simply connected.
• Punctured disk, annulus: have holes → multiply connected.
• Whole plane \(\mathbb{C}\): simply connected. But \(\mathbb{C}\setminus\{0\}\) is not.

5) Analytic takeaways

• Cauchy–Goursat on simply connected domains. If \(f\) is holomorphic on a simply connected domain, all closed contour integrals vanish: \[ \oint_{\gamma} f(z)\,dz = 0. \]
• Primitives exist. On a simply connected domain, every holomorphic function admits an antiderivative.
• Detecting holes. Nontrivial winding around a missing point (e.g., \(\mathbb{C}\setminus\{0\}\)) obstructs contraction.

6) Quick classification guide

• Open Disk: open, simply connected.
• Closed Disk: closed, simply connected.
• Annulus: open, multiply connected (one hole).
• Punctured Disk: open, multiply connected.
• Closed Rectangle: closed, simply connected.
• Whole Plane \(\mathbb{C}\): open and closed (clopen), simply connected.