More on T-closed sets

• Camargo, Javier ^[1] ; Macías, Sergio ^[2]
  1. [1] Universidad Industrial de Santander

     Universidad Industrial de Santander

     Colombia

     
  2. [2] Universidad Nacional Autónoma de México

     Universidad Nacional Autónoma de México

     México

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 27, Nº. 1, 2026
• Idioma: inglés
• DOI: 10.4995/agt.23481
• Enlaces
  • Texto completo
• Resumen

  • We consider properties of the diagonal of a continuum that are used later in the paper. We continue the study of T-closed subsets of a continuum X. We prove that for a continuum X, the statements: ΔX  is a nonblock subcontinuum of X2, ΔX is a shore subcontinuum of X2 and ΔX is not a strong centre of X2 are equivalent, this result answers in the negative Questions 35 and 36 and Question 38 ( i ∈ { 4,5 } ) of the paper "Diagonals on the edge of the square of a continuum, by A. Illanes, V. Martínez-de-la-Vega, J. M. Martínez-Montejano and D. Michalik". We also include an example, giving a negative answer to Question 1.2 of the paper"Concerning when F1 (X) is a continuum of colocal connectedness in hyperspaces and symmetric products, Colloquium Math., 160 (2020), 297-307'', by V. Martínez-de-la-Vega, J. M. Martínez-Montejano. We characterised the T-closed subcontinua of the square of the pseudo-arc. We prove that the T-closed sets of the product of two continua is compact if and only if such product is locally connected. We show that for a chainable continuum X, ΔX is a T-closed subcontinuum of X2 if and only if X is an arc. We prove that if X is a continuum with the property of Kelley, then the following are equivalent: ΔX is a T-closed subcontinuum of X2, X2\ΔX is strongly continuumwise connected, ΔX is a subcontinuum of colocal connectedness, and X2\ΔX is continuumwise connected. We give models for the families of T-closed sets and T-closed subcontinua of various families of continua.

• Referencias bibliográficas
  • D. P. Bellamy, Continua for which the set function T is continuous, Trans. Amer. Math. Soc. 151 (1970), 581-587. https://doi.org/10.2307/1995514
  • D. P. Bellamy and H. S. Davis, Continuum neighborhoods and filterbases, Proc. Amer. Math. Soc. 27 (1971), 371-374.10.1090/S0002-9939-1971-0276913-2
  • D. P. Bellamy, L. Fernández and S. Macías, On T-closed sets, Topology Appl. 195 (2015), 209-225. https://doi.org/10.1016/j.topol.2015.09.022
  • R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729-742. https://doi.org/10.1215/S0012-7094-48-01563-4
  • R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43-51. https://doi.org/10.2140/pjm.1951.1.43
  • R. H. Bing and F. B. Jones, Another homogeneous plane continuum, Trans. Amer. Math. Soc. 90 (1959), 171-192. https://doi.org/10.1090/S0002-9947-1959-0100823-3
  • J. Bobok, P. Pyrih, and V. Vejnar, On blockers in continua, Topol. Appl. 202 (2016), 346-355. https://doi.org/10.1016/j.topol.2016.01.013
  • J. Camargo and C. Uzcátegui, Continuity of the Jones' set function T, Proc. Amer. Math. Soc. 145 (2017), 893-899. https://doi.org/10.1090/proc/13379
  • F. Capulín, E. Castañeda-Alvarado, N. Ordóñez, and M. A. Ruiz, The hyperspace of T-closed subcontinua, Topology Appl. 275 (2020), 107154....
  • C. O. Christenson and W. L. Voxman, Aspects of Topology, Monographs and Textbooks in Pure and Applied Math., Vol. 39, Marcel Dekker, New York-Basel,...
  • D. W. Curtis and R. M. Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), 19-38. https://doi.org/10.4064/fm-101-1-19-38
  • H. S. Davis, A note on connectedness im kleinen, Proc. Amer. Math. Soc. 19 (1968), 1237-1241. https://doi.org/10.1090/S0002-9939-1968-0254814-3
  • H. S. Davis, Relationships between continuum neighborhoods in inverse limit sequences, Proc. Amer. Math. Soc. 64 (1977), 149-153.
  • H. S. Davis and P. H. Doyle, Invertible continua, Portugal. Math. 26 (1967), 487-491.
  • H. S. Davis, D. P. Stadtlander, and P. M. Swingle, Semigroups, continua and the set functions T^n, Duke Math. J. 29 (1962), 265-280
  • H. S. Davis, D. P. Stadtlander, and P. M. Swingle, Properties of the set functions T^n, Portugal. Math. 21 (1962), 113-133.
  • R. Duda, On the hyperspace of subcontinua of a finite graph, I, Fund. Math. 62 (1968), 265-286. https://doi.org/10.4064/fm-62-3-265-286
  • L. Fernández and S. Macías, The set functions T and K and irreducible continua, Colloq. Math. 121 (2010), 79-91. https://doi.org/10.4064/cm121-1-7
  • A. Illanes, V. Martínez-de-la-Vega, J. M. Martínez-Montejano and D. Michalik, Diagonals separating the square of a continuum, Bull. Malays....
  • A. Illanes, V. Martínez-de-la-Vega, J. M. Martínez-Montejano and D. Michalik, Diagonals on the edge of the square of a continuum, preprint.
  • F. B. Jones, Concerning Non-aposyndetic Continua, Amer. J. Math. 70 (1948), 403-413. https://doi.org/10.2307/2372339
  • H. Katsuura, Characterization of arcs by products and diagonals, Topology Proc. 63 (2024), 1-4.
  • K. Kuratowski, Topology, Vol. II, Academic Press, New York, 1968.
  • W. Lewis, Continuous Curves of Pseudo-arcs, Houston J. Math. 11 (1985), 91-99. https://doi.org/10.1016/0315-0860(85)90079-5
  • S. Macías, Aposyndetic properties of symmetric products of continua, Topology Proc. 22 (1997), 281-296.
  • S. Macías, A class of one-dimensional, nonlocally connected continua for which the set function T is continuous, Houston J. Math. 32 (2006),...
  • S. Macías, Homogeneous Continua for which the set function T is continuous, Topology Appl. 153 (2006), 3397-3401. https://doi.org/10.1016/j.topol.2006.02.003
  • S. Macías, A decomposition theorem for a class of continua for which the set function T is continuous, Colloq. Math. 109 (2007), 163-170.
  • S. Macías, On continuously irreducible continua, Topology Appl. 156 (2009), 2357-2363. https://doi.org/10.1016/j.topol.2009.01.024
  • S. Macías, On the idempotency of the set function T, Houston J. Math. 37 (2011), 1397-1305.
  • S. Macías, On continuously type A' theta-continua, JP Journal of Geometry and Topology 18 (2015), 1-14. https://doi.org/10.17654/JPGTAug2015_001_014
  • S. Macías, Topics on continua, 2nd Edition, Springer-Cham, 2018. https://doi.org/10.1007/978-3-319-90902-8
  • S. Macías, Set Function T: An Account on F. B. Jones' Contributions to Topology, Series Developments in Mathematics, Vol. 67, Springer,...
  • S. Macías, The set functions K and T and the property of Kelley, Topology Appl. 341 (2024), 108747. https://doi.org/10.1016/j.topol.2023.108747
  • S. Macías and S. B. Nadler, Jr., Various types of local connectedness in n-fold hyperspaces, Topology Appl. 154 (2007), 39-53. https://doi.org/10.1016/j.topol.2006.03.015
  • V. Martínez-de-la-Vega and J. Martínez-Montejano, Concerning when FF_1(X) is a continuum of colocal connectedness in hyperspaces and symmetric...
  • J. Martínez-Montejano, Mutual aposyndesis of symmetric products, Topology Proc. 24 (1999), 203-213.
  • E. E. Moise, An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua, Trans. Amer. Math. Soc. 63...
  • L. Mohler and L. G. Oversteegen, On the structure of tranches in continuously irreducible continua, Colloq. Math. 54 (1987), 23-28. https://doi.org/10.4064/cm-54-1-23-28
  • R. L. Moore, Foundations of point set theory, Rev. Ed., Amer. Math. Soc. Colloq. Publ., Vol. 13, Amer. Math. Soc., Providence, Rhode Island,...
  • S. B. Nadler, Jr., Hyperspaces of sets: A text with research questions, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49,...
  • S. B. Nadler, Jr., Continuum theory: An introduction, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 158, Marcel Dekker, New...
  • N. Ordóñez, C. Piceno, R. Quiñones-Estrella, and H. Villanueva, On T-ft sets and the hyperspace of T-closed sets, Topology Appl. 322 (2022),...
  • R. M. Schori and J. E. West, 2^I is homeomorphic to the Hilbert cube, Bull. Amer. Math. Soc. 78 (1972), 402-406. https://doi.org/10.1090/S0002-9904-1972-12917-3
  • R. M. Schori and J. E. West, Hyperspaces of graphs are Hilbert cubes, Pacific J. Math. 53 (1974), 239-251. https://doi.org/10.2140/pjm.1974.53.239
  • E. S. Thomas, Jr., Monotone decompositions of irreducible continua, Dissertationes Math. (Rozprawy Mat.) 50 (1966), 1-74.
  • R. Wardle, On a property of J. L. Kelley, Houston J. Math. 3 (1977), 291-299.
  • G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Publ., Vol. 28, Amer. Math. Soc., Providence, R. I., 1942.
  • S. Willard, General Topology, Addison-Wesley Publishing Company, Reading, Massachusetts, 1970.
  • E. J. Vought, Monotone decomposition of continua not separated by any subcontinua, Trans. Amer. Math. Soc. 192 (1974), 67-78.