Abstract
We prove basic results about the set function \({\mathcal {T}}\) defined by F. Burton Jones (Amer J Math 70:403–413, 1948) to study the properties of metric continua. We define this function on compacta, and then we concentrate on continua. In particular, we present some of the well known properties (such as connectedness im kleinen, local connectedness, semi-local connectedness, etc.) using the set function \({\mathcal {T}}\). The notion of aposyndesis was the main motivation of Jones to define this function. We present some properties of a continuum when it is \({\mathcal {T}}\)-symmetric and \({\mathcal {T}}\)-additive. We give properties of continuum on which \({\mathcal {T}}\) is idempotent, idempotent on closed sets and idempotent on contina. We also present results about the set functions \({\mathcal {T}}^n\), when \(n\in {\mathbb {N}}\).
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References
D. P. Bellamy, Continua for Which the Set Function \(\mathcal {T}\) is Continuous, Trans. Amer. Math. Soc., 1511 (1970), 581–587.
D. P. Bellamy, Set Functions and Continuous Maps, in General Topology and Modern Analysis, (L. F. McAuley and M. M. Rao, eds.), Academic Press, (1981), 31–38.
D. P. Bellamy, Some Topics in Modern Continua Theory, in Continua Decompositions Manifolds, (R H Bing, W. T. Eaton and M. P. Starbird, eds.), University of Texas Press, (1983), 1–26.
D. P. Bellamy and H. S. Davis, Continuum Neighborhoods and Filterbases, Proc. Amer. Math. Soc., 27 (1971), 371–374.
C. E. Burgess, Continua and Their Complementary Domains in the Plane, II, Duke Math. J., 19 (1952), 223–230.
C. E. Burgess, Continua Which are the Sum of a Finite Number of Indecomposable Continua, Proc. Amer. Math. Soc., 4 (1953), 234–239.
C. E. Burgess, Separation Properties and n-indecomposable Continua, Duke Math. J., 23 (1956), 595–599.
J. Camargo, S. Macías and M, Ruiz, Some Aspects Related to the Jones’ Set Function \(\mathcal {T}\), Topology Appl., 266 (2019), 106835.
J. Camargo, S. Macías, C. Uzcátegui, On the Images of Jones’ Set Function \(\mathcal {T}\), Colloquium Math., 153 (2018), 1–19.
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, 1977.
H. S. Davis, A Note on Connectedness im Kleinen, Proc. Amer. Math. Soc., 19 (1968), 1237–1241.
H. S. Davis, Relationships Between Continuum Neighborhoods in Inverse Limit Spaces and Separations 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, Properties of the Set Functions \(\mathcal {T}^n\), Portugal. Math., 21 (1962), 113–133.
H. S. Davis, D. P. Stadtlander and P. M. Swingle, Semigroups, Continua and the Set Functions \(\mathcal {T}^n\), Duke Math. J., 29 (1962), 265–280.
S. Gorka, Several Set Functions and Continuous Maps, Ph. D. Dissertation, University of Delaware, 1997.
J. G. Hocking and G. S. Young, Topology, Dover Publications, Inc., New York, 1988.
F. B. Jones, Concerning Nonaposyndetic Continua, Amer. J. Math., 70 (1948), 403–413.
S. Macías, On the Idempotency of the Set Function \(\mathcal {T}\), Houston J. Math., 37 (2011), 1297–1305.
S. Macías, Hausdorff Continua and the Uniform Property of Effros, Topology Appl., 230 (2017), 338–352.
S. Macías, Topics on Continua, 2nd Edition, Springer-Cham, 2018.
S. Macías and S. B. Nadler, Jr., On Hereditarily Decomposable Homogeneous Continua, Topology Proc., 34 (2009), 131–145.
A. Martínez Rodríguez, Propiedades de la función \(\mathcal {T}\) de Jones y otras funciones tipo conjunto, Master’s Thesis, Facultad de Ciencias, U. A. E. Mex., (2018) (Spanish)
M. A. Molina, Algunos Aspectos Sobre la Función \(\mathcal {T}\) de Jones, Bachelor’s Thesis, Facultad de Ciencias, U. N. A. M., 1998. (Spanish)
R. L. Moore, Foundations of Point Set Theory, Rev. Ed., Amer. Math. Soc. Colloq. Publ., Vol. 13, Amer. Math. Soc., Providence, Rhode Island, 1962.
D. Stadtlander, Further Properties of the Set Function \(\mathcal {T}\), J. Natur. Sci. and Math., 7 (1967), 91–94.
E. L. VandenBoss, Set Functions and Local Connectivity, Ph. D. Dissertation, Michigan State University 1970.
A. D. Wallace, The Position of C-sets in Semigroups, Proc. Amer. Math. Soc., 6 (1955), 639–642.
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Macías, S. (2021). The Set Function \({\mathcal {T}}\). In: Set Function T. Developments in Mathematics, vol 67. Springer, Cham. https://doi.org/10.1007/978-3-030-65081-0_2
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