Quotients of n-fold hyperspaces

Given a continuum X   and an integer n≥2n≥2, let Cn(X)Cn(X) be the n-fold hyperspace of X consisting of nonempty closed subsets of X with at most n   components. We consider the quotient space C1n(X)=Cn(X)/C1(X), with the quotient topology. We prove several of its properties. For example, we show that C1n(X) is unicoherent; if X   has the property of Kelley, then C1n(X) is contractible; dim⁡(Cn(X))=dim⁡(C1n(X)); C1n([0,1]) and C1n(S1) are both 2n-dimensional Cantor manifolds; X   is locally connected if and only if C1n(X) is locally connected.