I am having problems trying to prove the following statement:
Let $\Omega \subset \mathbb C$ be a region (i.e., an open, nonempty, connected subset of $\mathbb C$).
Prove that for all $z_0,z_1 \in \Omega$ there is a piecewise $C^1$ curve $\gamma$, such that $\gamma(0)=z_0$ and $\gamma(1)=z_1$.
I don't know how to prove this proposition, I would appreciate any hints.
A standard way of doing this is by showing that the set of points $z$ such that $z$ can be joined to any other point in $\Omega$ by a path is both a closed and an open subset of $\Omega$. Since $\Omega$ is assumed connected, this can only happen if...