A continuum X is a q-continuum -continuum, for some positive integer n) if for each subcontinuum K of X, we have that  only has a finite number of components  has at most n components). Following Professor E. J. Vought, we say that q-continuum -continuum) is of type A provided that it admits a monotone upper semicontinuous decomposition  whose quotient space is a finite graph, and it is of type  if, in addition, the elements of the decomposition have empty interior. A q-continuum -continuum) of type  for which the decomposition  is continuous is a continuously type  q-continuum -continuum). We characterize continuously type  q-continua as those q-continua of type A  for which the set function  is continuous. We prove that each continuously type  q-continuum is a -continuum for some positive integer n. We show that the n-fold symmetric product of a continuously type  q-continuum, for which the elements of the decomposition are nondegenerate, is a Z-set in both the hyperspace of closed subsets of the continuum and the n-fold hyperspace of that continuum.

continuous decomposition, continuously type  q-continuum, continuum, hyperspace, idempotency, Jones’ set function  q-continuum, -continuum, type A q-continuum, type  q-continuum, upper semicontinuous decomposition, weakly irreducible continuum, Whitney map, Z-set.