On w-unicoherence, top-irreducibility and Whitney levels near the top
Submitted: 2025-02-01
|Accepted: 2025-10-30
|Published: 2026-02-26
Copyright (c) 2025 José Gerardo Ahuatzi-Reyes, Norberto Ordoñez, Hugo Villanueva
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Downloads
Keywords:
Continuum, hyperspace, Whitney level, top-irreducible, w-unicoherent, pseudo-circular, pseudo-linear
Supporting agencies:
First author thanks Secretaría de Ciencia, Humanidades, Tecnología e Innovación (SECIHTI) for the financial support granted through the program "Estancias Posdoctorales por México Convocatoria 2023(1)"
Abstract:
Given a metric continuum X, we consider C(X) as the collection of all subcontinua of X. S. López introduced the concepts of pseudo-linearity and pseudo-circularity in order to characterize continua having a positive Whitney level that is an arc or a simple closed curve. In this paper, we introduce the concepts of w-unicoherence and top-irreducibility. We study the relations between these and the well-known and more naturally-related properties defined in continuum theory, and with the concepts of pseudo-linearity and pseudo-circularity. Moreover, by using these new concepts, we obtain a new characterization of continua which have a positive Whitney level that is an arc or a simple closed curve. Also, we provide necessary conditions for a continuum to have a positive Whitney level which is a simple n-od.
References:
E. Abo-Zeid, Some properties of Whitney continua, Topology Proceedings 3 (1978), 301-312.
R. H. Bing, Snake-like continua, Duke Mathematical Journal 18 (1951), 653-663. https://doi.org/10.1215/S0012-7094-51-01857-1 DOI: https://doi.org/10.1215/S0012-7094-51-01857-1
C. Eberhart and S.~B. Nadler Jr., Irreducible Whitney levels, Houston Journal of Mathematics 6, number 3 (1980), 355-363.
A. Illanes, A continuum having its hyperspaces not locally contractible at the top, Proceedings of the American Mathematical Society 111 (1991), 1177-1182. https://doi.org/10.1090/S0002-9939-1991-1037209-7 DOI: https://doi.org/10.2307/2048586
A. Illanes, S. Macías and W. Lewis (eds.), Continuum Theory, Lecture Notes in Pure and Applied Mathematics, vol. 230, Marcel Dekker Inc. (2002).
A. Illanes and S. B. Nadler Jr., Hyperspaces: Fundamentals and Recent Advances, Monographs and Textbooks in Pure and Applied Mathematics, vol. 126, Marcel Dekker Inc. (1999).
A. Illanes and I. Puga, Determining finite graphs by their large Whitney levels, Topology and Its Applications 60 (1994), 173-184. https://doi.org/10.1016/0166-8641(94)00011-5 DOI: https://doi.org/10.1016/0166-8641(94)00011-5
J. Krasinkiewicz, On the hyperspaces of hereditarily indecomposable continua, Fundamenta Mathematicae 84, no. 3 (1974), 175-186. https://doi.org/10.4064/fm-84-3-175-186 DOI: https://doi.org/10.4064/fm-84-3-175-186
J. Krasinkiewicz and S. B. Nadler Jr., Whitney properties, Fundamenta Mathematicae 98, no. 2 (1978), 165-180. https://doi.org/10.4064/fm-98-2-165-180 DOI: https://doi.org/10.4064/fm-98-2-165-180
K. Kuratowski, Topology, volume 2, Academic Press and PWN, New York, London and Warzawa (1968).
W. Lewis, Continuous curves of pseudo-arcs, Houston Journal of Mathematics 11 (1985), 91-99.
S. López, Hyperspaces locally 2-cell at the top, in: Continuum Theory (A. Illanes, S. Macías and W. Lewis, eds.) Lecture Notes in Pure and Appl. Math., vol. 230, Marcel Dekker Inc. (2002), 173-190. https://doi.org/10.1201/9780203910245.ch13 DOI: https://doi.org/10.1201/9780203910245.ch13
S. López and N. Ordoñez, Continua X for which C(X) has a plane neighborhood at the top, Topology Proceedings 44 (2014), 151-159.
L. Montejano-Peimbert and I. Puga-Espinosa, Shore points in dendroids and conical pointed hyperspaces, Topology and its Applications 46 (1992), 41-54. https://doi.org/10.1016/0166-8641(92)90038-2 DOI: https://doi.org/10.1016/0166-8641(92)90038-2
S.~B. Nadler Jr., Continuum Theory: An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker Inc. (1992).
S. B. Nadler Jr., Hyperspaces of Sets: A Text with Research Questions, Monographs and Textbooks in Pure and Applied Mathematics, vol. 49, Marcel Dekker Inc. (1978).
N. Ordoñez, Hyperspaces through regular and meager subcontinua, Topology and its Applications 300 (2021), 107760. https://doi.org/10.1016/j.topol.2021.107760 DOI: https://doi.org/10.1016/j.topol.2021.107760
I. Puga-Espinosa, Hiperespacios con punta de cono, Tesis doctoral, Facultad de Ciencias, Universidad Nacional Autónoma de México (1989).
