The connectedness of subsets in a continuum implies connectedness of Vietoric sets in the hyperspace $C_n(X)$
Keywords:
Continua, hyperspaces, Vietoris topologyAbstract
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 $X,\left\{C_1, \ldots, C_r\right\}$, it is well known that the set $\left\langle C_1, \ldots, C_r\right\rangle_n$ defined as the set 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 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 $r \leq n$, then $\left\langle C_1, \ldots, C_r\right\rangle \cap C_n(X)$ is a connected (arcwise connected) subspace of $C_n(X)$.
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