Increasing unions of starlike sets
Let S be a subset of \({\mathbb{R}}^ 2\). For points x in the closure of S (denoted \(cl S)\) and y in S, we say that x sees y via S if and only if the segment (x,y] lies in S. For points x,y in \(cl S,\) y is clearly visible from x via S if and only if there is some neighborhood N of y such that x sees via S each point of \(N\cap S\). The set S is called starlike if and only if there is some point p in cl S such that p sees via S each point of S. The author proves the following theorem: Let \(S\subset {\mathbb{R}}^ 2\). Every three points of \(cl S\) are clearly visible via S from a common point of \(cl S\) if and only if each bounded subset of S may be extended to a starlike set in S. When this occurs, set S is expressible as an increasing union of starlike sets.
scientific article; zbMATH DE number 3994425
scientific article