Problems on hyperspaces of continua, some answers

For a metric continuum X, let Cn(X) be the hyperspace of nonempty closed subsets of X with at most n components. The continuum X is k-mutually aposyndetic provided that given k distinct points of X, there are k mutually disjoint subcontinua of X, each containing one of the points in its interior. Answering some questions in the literature, in this paper: (a) given n, k >= 2, we show that for each continuum X, the hyperspace Cn(X) is k-mutually aposyndetic; (b) we answer two questions related to the continuity of the function that assigns to each arc in X its end-points; and (c) we prove that the property of being an n-od is a Whitney property. (c) 2022 Elsevier B.V. All rights reserved.