Open Domain is Connected iff it is Path-Connected

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

Then $D$ is connected $D$ is path-connected.

Necessary Condition
Complex Plane is Metric Space shows that $\C$ is topologically equivalent to the Euclidean space $\R^2$.

Continuous Image of Connected Space is Connected/Corollary 1 shows that connectedness is a topological property.

The result follows from Connected Open Subset of Euclidean Space is Path-Connected.

Sufficient Condition
The result follows from Path-Connected Space is Connected.