The hyperspace of non blockers of $F_1(X)$

TL;DR

This paper investigates the properties of the hyperspace of non-blockers in continua, proving the uniqueness of the simple closed curve in this context and addressing a question from prior research.

Contribution

It characterizes the hyperspace of non-blockers of $F_1(X)$ and proves the simple closed curve is uniquely identified by this hyperspace, answering an open question.

Findings

The simple closed curve is the unique continuum with $NB(F_1(X))=F_1(X)$.

Properties of the hyperspace $NB(F_1(X))$ are established.

The paper confirms a conjecture related to non-blockers in continua.

Abstract

A continuum is a compact connected metric space. A non-empty closed subset $B$ of a continuum $X$ does not block $x\in X\setminus B$ provided that the union of all subcontinua of $X$ containing $x$ and contained in $X\setminus B$ is dense in $X$. We denote the collection of all non-empty closed subset $B$ of $X$ such that $B$ does not block each element of $X\setminus B$ by $NB(F_1(X))$. In this paper we show some properties of the hyperspace $NB(F_1(X))$. Particularly, we prove that the simple closed curve is the unique continuum $X$ such that $NB(F_1(X))=F_1(X)$, given a positive answer to a question posed by Escobedo, Estrada-Obreg\'on and Villanueva in 2012.