Continua whose hyperspace of nonblockers of F1(X) is a continuum

Abstract

A continuum is a compact connected metric space. A non-empty closed subset B of a continuum X does not block x ∈ X \B provided that the union of all subcontinua of X containing x and contained in X \B is a dense subset of X. The collection of all non-empty closed subsets B of X such that B does not block each element of X \B is denoted by NB(F_1(X). In this paper, we find conditions under NB(F_1(X)) and the hyperspace of non-weak cut sets NWC(X) coincide and we exhibit a dendroid X for which NWC(X) is a non-empty proper subset of NB(F_1(X)). Also, we present geometric models for NB(F_1(X)); particularly, some of them give examples for a question posed by Escobedo, López and Villanueva in 2012. Finally, we prove that there exists a family of continua X such that the collection of hyperspaces NB(F_1(X)) is an uncountable incomparable family of continua.