Definition:Connected Domain (Complex Analysis)

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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 $\gamma: \left[{a \,.\,.\, b}\right] \to D$ and $\sigma: \left[{c \,.\,.\, d}\right] \to D$ be two such paths with $\gamma \left({a}\right) = \sigma \left({c}\right)$ and $\gamma \left({b}\right) = \sigma \left({d}\right)$.


Then $D$ is simply connected if and only if there exists a continuous function $H: \left[{0 \,.\,.\, 1}\right] \times \left[{0 \,.\,.\, 1}\right] \to D$ such that:

$H \left({t, 0}\right) = \gamma \left({t}\right)$ for all $t \in \left[{0 \,.\,.\, 1}\right]$.
$H \left({t, 1}\right) = \sigma \left({t}\right)$ for all $t \in \left[{0 \,.\,.\, 1}\right]$.

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.


Also see


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.


Sources