The hyperspace of non-blockers of singletons, all the possible examples

TL;DR

This paper explores the structure of non-blockers in metric continua, showing that for any completely metrizable separable space, there exists a continuum whose non-blocker collection is homeomorphic to it, answering several open questions.

Contribution

It constructs continua with non-blocker collections homeomorphic to any given completely metrizable separable space, advancing understanding of non-blocker topologies.

Findings

Existence of continua with prescribed non-blocker collections

Homeomorphism between non-blocker collections and given spaces

Resolution of open questions by Camargo et al.

Abstract

Given a metric continuum $X$, a nonempty proper closed subspace $B$ of $X$, does not block a point $p\in X\setminus B$ provided that the union of all subcontinua of $X$ containing $p$ and contained in $X\setminus B$ is a dense subset of $X$. The collection of all nonempty proper closed subspaces $B$ of $X$ such that $B$ does not block any element of $X\setminus B$ is denoted by $NB(F_{1}(X))$. In this paper we prove that for each completely metrizable and separable space $Z$, there exists a continuum $X$ such that $Z$ is homeomorphic to $NB(F_{1}(X))$. This answers a series of questions by Camargo, Capul\'in, Casta\=neda-Alvarado and Maya.