Arcwise connectedness of the complement in a hyperspace

Abstract

The hyperspace $C(X)$ of a continuum $X$ is always arcwise connected. In [6], S. B. Nadler Jr. and J. Quinn show that if $C(X)-\{A_{j}\}$ is arcwise connected for each $i=1,2$, then $C(X)-\{A_{1},A_{2}\}$ is also arcwise connected. Nadler raised questions in his book [5]: Is it still true with the two sets $A_{1}$ and $A_{2}$ replaced by $n$ sets, $n$ finite? What about countably many? What about a collection $\{A_{\lambda}:\lambda\in\Lambda\}$ which is a compact zero-dimensional subset of the hyperspace? In this paper we prove that if $\mathscr{A}\subset C(X)$ is a closed countable subset, $\mathscr{U}$ is an arc component of an open set of $C(X)$ and $C(X)-\{A\}$ is arcwise connected for each $A\in mathscr{A}$, then $\mathscr{U}-\mathscr{A}$ is arcwise connected.