Continuity of the Jones’ set function $\mathcal {T}$
HTML articles powered by AMS MathViewer
- by Javier Camargo and Carlos Uzcátegui PDF
- Proc. Amer. Math. Soc. 145 (2017), 893-899 Request permission
Abstract:
Given a continuum $X$, for each $A\subseteq X$, the Jones’ set function $\mathcal {T}$ is defined by $\mathcal {T}(A)=\{x\in X : \text {for each subcontinuum }K\text { such that }x\in \textrm {Int}(K), \text { then }K\cap A\neq \emptyset \}.$ We show that $\mathcal {D}=\{\mathcal {T}(\{x\}):x\in X\}$ is a decomposition of $X$ when $\mathcal {T}$ is continuous (restricted to the hyperspace $2^{X}$). We present a characterization of the continuity of $\mathcal {T}$ and answer several open questions posed by D. Bellamy.References
- David P. Bellamy, Continua for which the set function $T$ is continuous, Trans. Amer. Math. Soc. 151 (1970), 581–587. MR 271910, DOI 10.1090/S0002-9947-1970-0271910-9
- David P. Bellamy, Some topics in modern continua theory, Continua, decompositions, manifolds (Austin, Tex., 1980) Univ. Texas Press, Austin, Tex., 1983, pp. 1–26. MR 711974
- David P. Bellamy and Janusz J. Charatonik, The set function $T$ and contractibility of continua, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 1, 47–49 (English, with Russian summary). MR 500858
- David P. Bellamy, Leobardo Fernández, and Sergio Macías, On $\mathcal {T}$-closed sets, Topology Appl. 195 (2015), 209–225. MR 3414885, DOI 10.1016/j.topol.2015.09.022
- Howard Cook, W. T. Ingram, and Andrew Lelek, A list of problems known as Houston problem book, Continua (Cincinnati, OH, 1994) Lecture Notes in Pure and Appl. Math., vol. 170, Dekker, New York, 1995, pp. 365–398. MR 1326857, DOI 10.2307/3618329
- F. Burton Jones, Concerning non-aposyndetic continua, Amer. J. Math. 70 (1948), 403–413. MR 25161, DOI 10.2307/2372339
- Leobardo Fernández, On strictly point $\scr T$-asymmetric continua, Topology Proc. 35 (2010), 91–96. MR 2511797, DOI 10.4064/cm121-1-7
- Leobardo Fernández and Sergio Macías, The set functions $\scr T$ and $\scr K$ and irreducible continua, Colloq. Math. 121 (2010), no. 1, 79–91. MR 2725703, DOI 10.4064/cm121-1-7
- K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
- Sergio Macías, A class of one-dimensional, nonlocally connected continua for which the set function $\scr T$ is continuous, Houston J. Math. 32 (2006), no. 1, 161–165. MR 2202359
- Sergio Macías, Homogeneous continua for which the set function $\scr T$ is continuous, Topology Appl. 153 (2006), no. 18, 3397–3401. MR 2270591, DOI 10.1016/j.topol.2006.02.003
- Sergio Macías, On continuously irreducible continua, Topology Appl. 156 (2009), no. 14, 2357–2363. MR 2547780, DOI 10.1016/j.topol.2009.01.024
- Sergio Macías, A decomposition theorem for a class of continua for which the set function $\scr T$ is continuous, Colloq. Math. 109 (2007), no. 1, 163–170. MR 2308833, DOI 10.4064/cm109-1-13
- Sergio Macías, Topics on continua, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2147759, DOI 10.1201/9781420026535
- Sam B. Nadler Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49, Marcel Dekker, Inc., New York-Basel, 1978. A text with research questions. MR 0500811
- E. S. Thomas Jr., Monotone decompositions of irreducible continua, Rozprawy Mat. 50 (1966), 74. MR 196721
Additional Information
- Javier Camargo
- Affiliation: Escuela de Matemáticas, Facultad de Ciencias, Universidad Industrial de Santander, Ciudad Universitaria, Carrera 27 Calle 9, Bucaramanga, Santander, A.A. 678, Colombia
- Email: jcamargo@saber.uis.edu.co
- Carlos Uzcátegui
- Affiliation: Escuela de Matemáticas, Facultad de Ciencias, Universidad Industrial de Santander, Ciudad Universitaria, Carrera 27 Calle 9, Bucaramanga, Santander, A.A. 678, Colombia – and – Centro Interdisciplinario de Lógica y Álgebra, Facultad de Ciencias, Universidad de Los Andes, Mérida, Venezuela
- Email: cuzcatea@saber.uis.edu.co
- Received by editor(s): December 14, 2015
- Received by editor(s) in revised form: April 15, 2016
- Published electronically: October 3, 2016
- Additional Notes: The authors thank La Vicerrectoría de Investigación y Extensión de la Universidad Industrial de Santander for the financial support for this work, which is part of the VIE project #1873.
- Communicated by: Michael Wolf
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 893-899
- MSC (2010): Primary 54B20; Secondary 54C60
- DOI: https://doi.org/10.1090/proc/13379
- MathSciNet review: 3577889