Zero-dimensional subsets of hyperspaces

Abstract

Let $X$ be a metric continuum, let $2^{X}$ be the hyperspaces of all the nonempty closed subsets of $X$ and let $C(X)$ be the hyperspace of subcontinua of $X$. In this paper we prove: THEOREM 1. If $\mathscr{H}$ is a O-dimensional subset of $2^{X}$, then $2^{X}-\mathscr{H}$ is connected. THEOREM 2. If $\mathscr{H}$ is a closed O-dimensional subset of $C(X)$ such that $C(X)-\{A\}$ is arcwise connected for each $A\in \mathscr{H}$,then $C(X)-\mathscr{H}$ is arcwise connected. Theorem 2 answers a question by Sam B. Nadler, Jr.