The hyperspace of closed subsets on the edge of irreducible continua
Submitted: 2025-05-31
|Accepted: 2025-08-16
|Published: 2025-10-01
Copyright (c) 2025 Félix Capulín, David Maya, Fernando Orozco-Zitli, Emiliano Rodríguez-Anaya
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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Keywords:
irreducible continuum, non-weak cut set, non-block set, weak non-block set, shore set, non-cut set
Supporting agencies:
Consejo Nacional de Ciencia y Tecnología
Abstract:
For a nonempty nowhere dense closed subset of a continuum $X$, consider the following properties: being a non-weak cut subset, a non-block subset, a weak non-block subset, a shore subset, a non-strong center, and a non-cut subset of X. In this paper, we provide necessary conditions for subsets of an irreducible continuum about a subset to have one of these properties, and we prove that these properties are equivalent for nonempty nowhere dense closed subsets of an irreducible continuum about a finite subset. This result completes the study previously conducted on non-cut points and for subsets on the edge of a continuum by several authors.
References:
J. G. Anaya and D. Maya, Non-cut ordered arcs of the hyperspace of subcontinua, Topology Appl. 349 (2024), 108908. https://doi.org/10.1016/j.topol.2024.108908
J. Bobok, P. Pyrih, and B. Vejnar, Non-cut, shore and non-block points in continua, Glas. Mat. Ser. III 1, no. 51 (2016), 237-253. https://doi.org/10.3336/gm.51.1.14
J. Camargo, F. Capulín, E. Castañeda-Alvarado, and D. Maya, Continua whose hyperspace of nonblockers of F1(X) is a continuum, Topology Appl. 262 (2019), 30-40. https://doi.org/10.1016/j.topol.2019.05.007
J. Camargo and M. Ferreira, Nonblockers for hereditarily decomposable continua with the property of Kelley, Topology Appl. 342 (2024), 108782. https://doi.org/10.1016/j.topol.2023.108782
J. Camargo, S. Macías, and D. Maya, C1(X) on the edge of C2(X), Appl. Gen. Topol. 26, no. 1 (2025), 341-368. https://doi.org/10.4995/agt.2025.22003
J. Camargo, D. Maya, and L. Ortiz, The hyperspace of nonblockers of F1(X), Topology Appl. 251 (2019), 70-81. https://doi.org/10.1016/j.topol.2018.10.007
J. Camargo, D. Maya, and P. Pellicer-Covarrubias, Noncut subsets of the hyperspace of subcontinua, Topology Appl. 305 (2022), 107867. https://doi.org/10.1016/j.topol.2021.107867
R. Escobedo, C. Estrada-Obregón, and J. Sánchez-Martínez, On hyperspaces of non-cut sets of continua, Topology Appl. 217 (2017), 97-106. https://doi.org/10.1016/j.topol.2016.12.004
D. Harper, The internal structure of irreducible continua, PhD Thesis, University of Oxford, 2017.
A. Illanes, V. Martínez de-la Vega, and J. Vega, The hyperspace of non-cut subcontinua of graphs, Topology Appl. 373 (2025), 109533. https://doi.org/10.1016/j.topol.2025.109533
A. Illanes and P. Krupski, Blockers in hyperspaces, Topology Appl. 158, no. 5 (2011), 653-659. https://doi.org/10.1016/j.topol.2011.01.001
A. Illanes, V. Martínez-de-la Vega, J. M. Martínez-Montejano, and D. Michalik, Diagonals of the square of a continuum, Topology Appl. 364 (2025), 109099.
https://doi.org/10.1016/j.topol.2024.109099
A. Illanes and S. B. Nadler, Jr., Hyperspaces: fundamentals and recent advances, Pure Appl. Math., Marcel Dekker, vol. 216, New York, NY: Marcel Dekker, 1999.
R. Leonel, Shore points of a continuum, Topology Appl. 161 (2014), 433-441. https://doi.org/10.1016/j.topol.2013.11.004
S. Macías, Non-blockers of decomposable continua with the property of Kelley, and the set function T, Topol. Proc. 63 (2024), 29-38.
V. Martínez-de-la Vega and J. M. Martínez-Montejano, Concerning when F1(X) is a continuum of colocal connectedness in hyperspaces and symmetric products, Colloq. Math. 160, no. 2 (2020), 297-307. https://doi.org/10.4064/cm7611-3-2019
S. B. Nadler, Jr., Continuum theory. An introduction, Pure Appl. Math., Marcel Dekker, vol. 158, New York: Marcel, 1992.
S. Willard, General topology, Addison-Wesley Series in Mathematics. Reading, Mass. etc.: Addison-Wesley Publishing Company, 1970.
