The hyperspace of non-blockers of singletons, all the possible examples
Keywords:
Blocker, continuum, hyperspace, pseudo-arcAbstract
Given a metric continuum $X$, a nonempty proper closed subspace $B$ of $X$, does not block a point $p \in X \backslash B$ provided that the union of all subcontinua of $X$ containing $p$ and contained in $X \backslash B$ 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 $X \backslash B$ is denoted by $N B\left(F_1(X)\right)$. 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 $N B\left(F_1(X)\right)$. This answers a series of questions by Camargo, Capulín, Castan̄eda-Alvarado and Maya.
References
J. Bobok, P. Pyrih, B. Vejnar, Non-cut, shore and non-block points in continua, Glas. Mat. Ser. III 51 (71) (2016), 237-253.
J. Bobok, P. Pyrih, B. Vejnar, On blockers in continua, Topology Appl. 202 (2016), 346-355.
J. Camargo, F. Capulín, E. Castan̄eda-Alvarado, D. Maya Continua whose hyperspace of nonblockers of $\mathcal{F}_1(X)$ is a continuum, Topology Appl. 262 (2019), 30-40.
J. Camargo, D. Maya, L. Ortiz, The hyperspace of nonblockers of $\mathcal{F}_1(X)$, Topology Appl. 251 (2019), 70-81.
R. Escobedo, M. de Jesús López, H. Villanueva, Nonblockers in hyperspaces, Topology Appl. 159 (2012), 3612-3618.
A. Illanes, P. Krupski, Blockers in hyperspaces, Topology Appl. 158 (2011), 653659.
A. Illanes and S. B. Nadler, Jr., Hyperspaces, Fundamentals and recent advances, Monographs and Textbooks in Pure and Applied Math. Vol. 216, Marcel Dekker, Inc. New York and Basel, 1999.
W. Lewis, The pseudo-arc, Bol. Soc. Mat. Mexicana (3) 5 (1999), 25-77.
S. B. Nadler, Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Math. Vol. 49, Marcel Dekker, Inc., New York, N. Y., 1978.
S. B. Nadler, Jr., Continuum Theory, An introduction, Monographs and Textbooks in Pure and Applied Math. Vol. 158, Marcel Dekker, Inc. New York, N.Y., 1992.
C. Piceno, Nonblockers in homogeneous continua, Topology Appl. 249 (2018) 127-134.
