Properties of the (n, m)−fold hyperspace suspension of continua

Let n, m ∈ N with m ≤ n and X be a metric continuum. We consider the hyperspaces Cn(X) (respectively, Fn(X)) of all nonempty closed subsets of X with at most n components (respectively, n points). The (n, m)−fold hyperspace suspension on X was introduced in 2018 by Anaya, Maya, and Vázquez-Juárez, to be the quotient space Cn(X)/Fm(X) which is obtained from Cn(X) by identifying Fm(X) into a one-point set. In this paper we prove that Cn(X)/Fm(X) contains an n−cell; Cn(X)/Fm(X) has property (b); Cn(X)/Fm(X) is unicoherent; Cn(X)/Fm(X) is colocally connected; Cn(X)/Fm(X) is aposyndetic; and Cn(X)/Fm(X) is finitely aposyndetic.