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

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 NB(F-1(X )) of 2X consists of all elements A is an element of 2(X) - {X} such that for each x is an element of 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 NB(F-1(X )) 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. Capulin, E. Castaneda-Alvarado and D. Maya. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.