Certain theorems that apply to
compact, metric continua that are separated by none of their subcontinua can be
generalized and strengthened in those continua that are separated by none of their
nonaposyndetic subcontinua. For those of the former type, if the continuum
is aposyndetic at a point, it is locally connected at the point. The same
conclusion is possible if the continuum is not separated by any nonaposyndetic
subcontinuum. Also, if a continuum is separated by no subcontinuum and cut by no
point, it is a simple closed curve. A second result of this paper is to prove
that if no nonaposyndetic subcontinuum separates and no point cuts the
continuum, then it is a cyclically connected continuous curve; in fact this yields
a characterization of hereditarily locally connected, cyclically connected
continua.
A third theorem characterizes an hereditarily locally connected continuum as an
aposyndetic continuum that is separated by no nonaposyndetic subcontinuum. This
is a somewhat stronger result than the known equivalence of hereditary local
connectedness and hereditary aposyndesis.
