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

Definition
Let $D \subseteq \C$ be a subset of the set of complex numbers. $D$ is called a simply connected domain iff $D$ is open and simply connected.

Simply connected requirements
For $D$ to be simply connected, it is required that all paths in $D$ are homotopic.

This implies that if $\gamma, \sigma: \left[{0 \,.\,.\, 1}\right] \to D$ are two paths in $D$, then 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.