The hyperspace of nonblockers of F1(X)

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 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 \ B by N B(F 1 (X)). In this paper we show some properties of the hyperspace N B(F 1 (X)). Particularly, we prove that the simple closed curve is the unique continuum X such that N B(F 1 (X)) = F 1 (X), given a positive answer to a question posed by Escobedo, Estrada-Obregón and Villanueva in 2012.