Nonblockers for hereditarily decomposable continua with the property of Kelley

Abstract

Given a continuum $X$, let $\mathcal{N}\mathcal{B}({\mathcal{F}}_1(X))$ be the hyperspace of nonblockers of ${\mathcal{F}}_1(X)$. In this paper, we show that if $X$ is hereditarily decomposable with the property of Kelley such that $\mathcal{N}\mathcal{B}({\mathcal{F}}_1(X))$ is a continuum, then $X$ is a simple closed curve. Thus, we characterize the simple closed curve as the unique hereditarily decomposable continuum with the property of Kelley $X$ such that its hyperspace $\mathcal{N}\mathcal{B}({\mathcal{F}}_1(X))$ is a continuum.