◀ ▲ ▶Branches / Analysis / Definition: Interior, Boundary, and Closures of a Region in the Complex Plane
Definition: Interior, Boundary, and Closures of a Region in the Complex Plane
Let $U\subseteq\mathbb C$ be a subset of the complex plane.
* A point $b\in U$ is said to be in the interior of $U$, if there is a disc $B(b,r)$ contained in $U.$
* A point $b$ is called a boundary point of $U,$ if every disc centered at $b$ contains both, the points of the interior of $U$ and outside the interior of $U.$
* The closure of $U$ is the set union of $U$ with its boundary points.
Note
 The above definitions are specific instances of the general definitions in topology, which can be found here.
 With this respect, the complex plane is a specific topological space together with absolute value as a distance measure.
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References
Bibliography
 Riesz F., Sz.Nagy, B.: "Functional Analysis (Translation)", Dover Publications, 1955
 Kneser, Hellmuth: "Mathematische Lehrbücher  Funktionentheorie", Vanderhoeck & Ruprecht, 1958
 Lang, Serge: "Complex Analysis", Springer, 1999, Forth Edition