Non-blockers of decomposable continua with the property of Kelley, and the set function $\mathcal{T}$

Abstract

Given a continuum $X$, let $\mathcal{F}_1(X)$ be the family of singletons of $X$ and let $\mathcal{N} \mathcal{B}\left(\mathcal{F}_1(X)\right)$ be the hyperspace of non-blockers of $\mathcal{F}_1(X)$. We use Professor Jones' set function $\mathcal{T}$ to present a different proof of the two main theorems of J. Camargo and M. Ferreira in Nonblockers for hereditarily decomposable continua with the property of Kelley. We show a proposition of independent interest, which is key to present a different proof of the second main theorem of Camargo and Ferreira, namely: if $X$ is a hereditarily decomposable continuum with the property of Kelley such that $\mathcal{N B}\left(\mathcal{F}_1(X)\right)$ is a continuum, then $X$ is a simple closed curve. We also prove that if $X$ is a decomposable, not hereditarily decomposable continuum with the property of Kelley and $\mathcal{N B}\left(\mathcal{F}_1(X)\right)$ is a continuum, then the set function $\mathcal{T}$ is continuous. Hence, $X$ has a continuous decomposition $\mathcal{G}$ such that many elements of $\mathcal{G}$ are indecomposable subcontinua of $X$.