Let M be a compact
metric continuum. If x ∈ M let K(x) be the set to which an element y of M
belongs if and only if M is not aposyndetic at x wilh respect to y. If, for all
x in M,x ∈Int(K(x)), then M is the union of a countable collection of
indecomposable subcontinua each of which is the closure of an open set. There exists
a compact metric continuum M and a dense subset J of M such that for
each x ∈ J,K(x) = S, but M contains no indecomposable subcontinua with
nonvoid interior. It is the case, however, that if M has a point x such that Int
(K(x)) = ∅ lhen M contains an indecomposable subcontinua which intersects Int
(K(x)).
