The notion of terminal
subcontinuum of a continuum is introduced as a generalization of the idea of terminal
point and is used to study the structure of a large class ℳ of hereditarily unicoherent
Hausdorff continua. The class ℳ contains all hereditarily unicoherent metric
continua and all hereditarily decomposable, hereditarily unicoherent Hausdorff
continua. The major result is that every member of ℳ is irreducible about the union
of its indecomposable terminal subcontinua. The known result that a hereditarily
decomposable, hereditarily unicoherent Hausdorff continuum is irreducible about its
terminal points is a corollary.
