If a point set is both connected
and closed it is called a continuum. The structure of a nonlocally connected
continuum can be described in terms of its aposyndetic properties. In this
paper various forms of continuum aposyndesis, that is, aposyndesis with
respect to subcontinua, are considered. It is shown that the presence of any of
these forms of aposyndesis in a compact metric continuum which is totally
nonconnected im kleinen (not connected im kleinen at any point) insures
nonsemi-local-connectedness on a dense open subset of the continuum and the set of
weak cut points in each open subset of the continuum has cardinality at least c1
A weak cut point theorem for compact plane continua is established. An
example is given which indicates that this result does not hold in Euclidean
3-space. Near aposyndesis, a generalization of aposyndesis, is introduced. It
is shown that the presence of this property in a totally nonaposyndetic,
separable, metric continuum implies the existence of uncountably many weak cut
points.
