Concerning k-mutual aposyndesis in symmetric products

Given a metric continuum X, let Fn(X) denote the hyperspace of nonempty subsets of X with at most n points. We prove that Fn(X) is k-mutually aposyndetic for every k⩾2 and n⩾3. That is, given k distinct elements in Fn(X), there are k disjoint subcontinua of Fn(X), each containing one of the elements in its interior.