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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 $U$.

Notes:

• $U$ is an open subset of the specific topological space which is the complex plane, together with absolute value as a distance measure.
• Illustration

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References

Bibliography

1. Lang, Serge: "Algebra - Graduate Texts in Mathematics", Springer, 2002, 3rd Edition
2. Kneser, Hellmuth: "Mathematische Lehrbücher - Funktionentheorie", Vanderhoeck & Ruprecht, 1958