Open Domain is Connected iff it is Path-Connected
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Theorem
Let $D \subseteq \C$ be a open subset of the set of complex numbers.
Then $D$ is connected if and only if $D$ is path-connected.
Proof
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.
$\Box$
Sufficient Condition
The result follows from Path-Connected Space is Connected.
$\blacksquare$