A note on continuous collections of disjoint continua

Abstract

Hamstrom has shown that if G is a continuous collection of disjoint arcs filling up a compact continuous curve M in the plane such that M/G is an arc, then G* (x e G* if and only if for some geG,xg) is a simple closed curve plus its interior. One purpose of this note is to show that if S is a space satisfying Axioms 0 -5 of R. L. Moore's Foundations of Point Set Theory, and M a S such that (1) M has one and only one complementary domain, and (2) there exists a continuous collection of disjoint nondegenerate continua filling up M, then M is a simple closed curve J plus one of the complementary domains of /. Another purpose of this note is to state and prove some consequences of this theorem.