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Let $\Omega \subset \mathbb{C}$ be an open set that is convex. Show that $\Omega=\bigcap H_i$, where $\Omega \subset H_i$ and $H_i = \{ z: Im(a_iz+b_i)>0, a_i,b_i \in \mathbb{C}\}$

It is pretty clear that $\Omega \subseteq \bigcap H_i$ by definition. But I am not sure how to show the converse. Any thought?

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It suffices to show that for any $p \notin \Omega$, there is a half plane $H$ containing $\Omega$ but not $p$. For convenience, translate so $p = 0$. Let $S$ be the set of $w$ with $|w| = 1$ such that the ray $\{t w: t > 0\}$ intersects $\Omega$. Show that $S$ and $-S = \{-w: w \in S\}$ are disjoint and open, and use the fact that the unit circle is connected...

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    To make the hint more clear: assume $p\in \bigcap H_i \setminus \Omega$. Then, there is a half plane $H \supseteq \Omega$ but $p\notin H$. A contradiction. – 2017-01-16
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    Thank you very much! But how do I show that $S$ and $-S$ are open? – 2017-01-16