Non-cut vietoric sets in n-fold hyperspaces of continua - Corona-Vázquez, Florencio; Martinez-Cortez, José; Quiñones-Estrella, Russell-Aarón; Sánchez-Martínez, Javier (Topology Proceedings, 2025)
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, C_n(X) is endowed with the Vietoris topology. Given a finite family {C_1, …, C_r} of connected subsets of X, r ≤ n, it is well known that the set C_1, …, C_r_n 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 connected subset of C_n(X), consequently, if B_1, …, B_r X are such that X-B_i is a connected subset for each i {1, …, r}, then C_n(X)-X-B_1, …, X-B_r_n has a connected complement in C_n(X). In this paper we will study the analogous property by changing non-cut sets for some of the following types of sets: non-weak cut sets, sets that do not block the singletons of X, sets that do not block some point of X, and shore sets.