Nonblockers for hereditarily decomposable continua with the property of Kelley

TL;DR

This paper characterizes the simple closed curve as the unique hereditarily decomposable continuum with the Kelley property for which the hyperspace of nonblockers is a continuum.

Contribution

It provides a characterization of simple closed curves via hyperspaces of nonblockers in hereditarily decomposable continua with the Kelley property.

Findings

If the hyperspace of nonblockers is a continuum, then the continuum is a simple closed curve.

The simple closed curve is uniquely characterized among hereditarily decomposable continua with the Kelley property.

The paper establishes a new link between hyperspace properties and the topology of the continuum.

Abstract

Given a continuum $X$, let $\mathcal{NB} (\mathcal{F}_1(X))$ 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 $\mathcal{NB} (\mathcal{F}_1(X))$ 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 $\mathcal{NB} (\mathcal{F}_1(X))$ is a continuum.