Let X be a compact
connected metric space and 2X(C(X)) denote the hyperspace of closed subsets
(subcontinua) of X. In this paper the hyperspaces are investigated with respect to
the property of aposyndesis. The main result states that each of 2X and
C(X) is aposyndetic. If X is semi-aposyndetic, then each of 2X and C(X)
is mutually aposyndetic. An example is given of a non-semi⋅aposyndetic
continuum for which C(X) is not mutually aposyndetic. In an extension
of the main result for C(X) it is shown that C(X) is countable closed set
aposyndetic. The techniques utilize the partially ordered structure of 2X and
C(X).
