Non-cut vietoric sets in $n$-fold hyperspaces of continua

Abstract

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 $\left\{C_1, \ldots, C_r\right\}$ of connected subsets of $X, r \leq n$, it is well known that the set $\left\langle C_1, \ldots, C_r\right\rangle_n$ of all elements $A$ in $C_n(X)$ such that $A \subset \bigcup_{i=1}^r C_i$ and $A \cap C_i \neq \emptyset$ for each $i$, is a connected subset of $C_n(X)$, consequently, if $B_1, \cdots, B_r \subset X$ are such that $X-B_i$ is a connected subset for each $i \in\{1, \ldots, r\}$, then $C_n(X)-\left\langle X-B_1, \ldots, X-B_r\right\rangle_n$ 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.