Requirement for Connected Domain to be Simply Connected Domain

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 $\gamma: \left[{a \,.\,.\, b}\right] \to D$ 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 iff 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.