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 $\R^2$.

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.

$\blacksquare$