Minima Nonblockers and Blocked Sets of a Continuum

Abstract

Given a continuum $X$ and an element $x \in X$, $\pi(x)$ is the smallest set that contains $x$ and does not block singletons, and $B(x)$ is the set of all elements blocked by ${x}$. We prove that for each $x \in X$, $B(x)$ is connected, $B(x) \subset \pi(x)$, and that if $B(x)$ is closed, then $B(x)=\pi(x)$. Among other results, we prove that if $X$ is a Kelley continuum and $\pi(x)$ is proper, then $B(x)=\pi(x)$. Finally, we prove that for a certain class of dendroids, the family of minima non-blockers coincides with the family of connected non-blockers.