Definition:Connected Domain (Complex Analysis)/Simply Connected Domain/Requirement

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Simply Connectedness Requirement

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

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.


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