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 » is a positive integer, the continuum M is said to be indecomposable under index » if If is the finished sum of « continua and is not the finished sum of »+1 continua. Swingle has shown [7, Theorem 2] that if » is a positive integer and the continuum M is indecomposable under index w, then M is the finished sum of « indecomposable continua. The author has shown [2, Theorem l] that if « = 2 and the continuum M is indecomposable under index », and G is a collection of » 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 «. Also, there is given a necessary and sufficient condition that a compact continuum be indecomposable under index «. An indecomposable continuum can be described as a nondegenerate continuum which is indecomposable under index 1. If « = 1, then in order that a continuum M be indecomposable under index «, it is necessary and sufficient that M contain «+2 points such that M is irreducible about any « + 1 of them.* 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