A Property of Indecomposable Connected Sets

Introduction. Two nonempty subsets of a topological space are said to be separated if neither intersects the closure of the other. A set is connected if it is not the union of two separated sets. A connected set I is indecomposable if it is not the union of two connected sets, neither of which is dense in I. In [l], Swingle raised the following question: does there exist, in the plane, an indecomposable connected set I, such that the set IVJ {p\ fails to be indecomposable for some limit point p of I? The purpose of this paper is to prove that the answer is negative. It is interesting to note that the plane plays an essential role here as the embedding space: if the plane is replaced by Euclidean 3-space in Swingle's question, the answer turns out to be affirmative. The construction of such an example is rather complicated, and is not included in this paper. Notation. A component of a set is a maximal connected subset. If X and Y are sets, X—Y denotes the set of all elements of X which are not elements of Y (whether Y is a subset of X or not). The boundary of a set X will be denoted by dX, and X will denote the closure of X.