Metrizable spaces homeomorphic to the hyperspace of nonblockers of singletons of a continuum

Abstract

A continuum is a nondegenerate compact connected metric space. The hyperspace of all nonempty closed subsets of a continuum X topologized by the Hausdorff metric is denoted by $2^X$. Given a continuum X, the subspace $\mathrm{NB}\left(F_1\right(X\left)\right)$ of $2^X$ consists of all elements $A\in 2^X-\left\{X\right\}$ such that for each $x\in X-A$, the union of all subcontinua of X containing x and contained in $X-A$ is a dense subset of X. The members of $\mathrm{NB}\left(F_1\right(X\left)\right)$ are called nonblocker subsets of the singletons of the continuum X. In this paper, we show that each proper nonempty open subset U of a compact metric space can be embedded in a continuum X such that U and the hyperspace of nonblocker subsets of X are homeomorphic. This answers a question posed by J. Camargo, F. Capulín, E. Castañeda-Alvarado and D. Maya.