The connectedness of subsets in a continuum implies connectedness of Vietoric sets in the hyperspace C_n(X) - Corona-Vázquez, Florencio; Martínez-Cortez, José; Quiñones-Estrella, Russell-Aarón; Sánchez-Martinez, Javier (Topology Proceedings, 2024)
Let X be a continuum and n be a positive integer. The symbol C_n(X) denotes the hyperspace of all nonempty, closed subsets of X having at most n components, equipped with the Vietoris topology. Given a finite family of subcontinua of X,{C_1, …, C_r}, it is well known that the set C_1, …, C_r_n defined as the set of all elements A in C_n(X) such that A _i=1^r C_i and A C_i eq for each i, is a subcontinuum of C_n(X). In this paper, we extend the previous result by showing that, if each C_i is connected (arcwise connected) and r ≤ n, then C_1, …, C_rC_n(X) is a connected (arcwise connected) subspace of C_n(X).