Definition:Connected Domain (Complex Analysis)

Definition

Let $D \subseteq \C$ be a subset of the set of complex numbers.

Then $D$ is a connected domain if and only if $D$ is open and connected.

Simply Connected Domain

Let $D \subseteq \C$ be a connected domain.

Then $D$ is called a simply connected domain if and only if $D$ is simply connected.

Simply Connectedness Requirement

For $D$ to be simply connected, it is required that two paths $\gamma, \sigma$ in $D$ with the same initial points and final points are freely homotopic.

Let $\gamma: \closedint a b \to D$ and $\sigma: \closedint c d \to D$ be two such paths with $\map \gamma a = \map \sigma c$ and $\map \gamma b = \map \sigma d$.

Then $D$ is simply connected if and only if there exists a continuous function $H: \closedint 0 1 \times \closedint 0 1 \to D$ such that:

$\map H {t, 0} = \map \gamma t$ for all $t \in \closedint 0 1$
$\map H {t, 1} = \map \sigma t$ for all $t \in \closedint 0 1$.

The function $H$ is called a (free) homotopy.

It follows from Homotopy Characterisation of Simply Connected Sets that this definition of a simply connected set is equal to the standard definition of a simply connected set.

Also known as

Some texts omit the word connected and simply call $D$ a domain.

Notes

A connected domain $D$ is often used as the domain of a complex-differentiable function $f: D \to \C$.

Then $D$ must be open by definition of complex-differentiable.

Also, connected sets that are not open may not be path-connected.

This is why the definition of connected domain requires that $D$ is open.