Continua which are the sum of a finite number of indecomposable continua

Swingle [7]1 has given the following definitions. (1) A continuum M is said to be the finished sum of the continua of a collection G if G = M and no continuum of G is a subset of the sum of the others.2 (2) If n is a positive integer, the continuum M is said to be indecomposable under index n if M is the finished sum of n continua and is not the finished sum of n +1 continua. Swingle has shown [7, Theorem 2] that if n is a positive integer and the continuum M is indecomposable under index n, then M is the finished sum of n indecomposable continua. The author has shown [2, Theorem 1] that if n = 2 and the continuum M is indecomposable under index n, and G is a collection of n indecomposable continua whose finished sum is M, then G is the only such collection. In the present paper, it is shown that for a compact continuum, this theorem holds for any positive integer n. Also, there is given a necessary and sufficient condition that a compact continuum be indecomposable under 'index n. An indecomposable continuum can be described as a nondegenerate continuum which is indecomposable under index 1. If n =1, then in order that a continuum M be indecomposable under index n, it is necessary and sufficient that M contain n+2 points such that M is irreducible about any n+1 of them.3 Swingle [7] has shown that it is impossible, in a certain manner, to generalize this theorem. Theorem 3 of the present paper might be considered a generalization of the necessary condition of the above theorem. However, it is easily seen that the converse of Theorem 3 is not true. Theorems 1-5 are proved on the basis of R. L. Moore's Axioms 0 and 18. Hence these theorems hold in any metric space.4