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 ∈ G∖∗ if 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.
is a simple closed curve plus its interior. One
purpose of this note is to show that if 