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Definition: Closed and Open Regions of the Complex Plane
Let $U\subseteq\mathbb C$ be a subset of the complex plane. We call $U$ an open region if for every point $u\in U$ there is a disc $D(u,r)$ centered at $u$ and some radius $r > 0$ such that the entire disc $D(u,r)$ is contained in $U$.
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References
Bibliography
 Lang, Serge: "Algebra  Graduate Texts in Mathematics", Springer, 2002, 3rd Edition
 Kneser, Hellmuth: "Mathematische Lehrbücher  Funktionentheorie", Vanderhoeck & Ruprecht, 1958