Concerning k-mutual aposyndesis in symmetric products

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