On the imbedding of a finite family of closed disks into a plane or S^{2}

Abstract

Let $\{V_{i}\}_{i=1}^{n}$ be a finite family of closed subsets of a plane or a sphere $S^{2}$, each homeomorphic to the two-dimensional disk. In this paper we discuss the question how the boundary of connected components of a complement $\rr^{2} \setminus \bigcup_{i=1}^{n} V_{i}$ (accordingly, $S^{2} \setminus \bigcup_{i=1}^{n} V_{i}$) is arranged. It appears, if a set $\bigcup_{i=1}^{n} \Int V_{i}$ is connected, that the boundary $\partial W$ of every connected component $W$ of the set $\rr^{2} \setminus \bigcup_{i=1}^{n} V_{i}$ (accordingly, $S^{2} \setminus \bigcup_{i=1}^{n} V_{i}$) is homeomorphic to a circle.