Vol. 20, No. 1, 1967

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A note on continuous collections of disjoint continua

Edward Lee Bethel

Vol. 20 (1967), No. 1, 21–30

M. E. 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 Gif and only if for some g G,x g) 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 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 J. Another purpose of this note is to state and prove some consequences of this theorem.

Mathematical Subject Classification
Primary: 54.55
Received: 16 December 1964
Published: 1 January 1967
Edward Lee Bethel