The Property of Semi-Kelley for Hausdorff Continua
Keywords:
confluent mapping, continuum, hyperspace, property of Kelley, property of semi-Kelley, retraction, semi-confluent mappingAbstract
In this paper we introduce the property of semi-Kelley for Hausdorff continua. We use this notion to characterize Hausdorff continua with the property of Kelley. We prove that if a product of Hausdorff continua has the property of semi-Kelley, then each factor continuum has the property of Kelley. Concerning hyperspaces, we prove that if either $C(X)$, $C_n(X)$, or $2^X$ has the property of semi-Kelley, then $X$ has the property of Kelley.
References
Enrique Castañeda-Alvarado and Ivon Vidal-Escobar, Property of being semi-Kelley for the cartesian products and hyperspaces, Comment. Math. Univ. Carolin. 58 (2017), no. 3, 359-369.
Mauricio Chacón-Tirado and María de J. López, On the property of Kelley for Hausdorff continua, Rev. Integr. Temas Mat. 38 (2020), no. 1, 55-66.
Janusz J. Charatonik and Wªodzimierz J. Charatonik, A weaker form of the property of Kelley, Topology Proc. 23 (1998), Spring, 69-99.
Janusz J. Charatonik and Wªodzimierz J. Charatonik, Whitney maps--a non-metric case, Colloq. Math. 83 (2000), no. 2, 305-307.
Janusz J. Charatonik and Alejandro Illanes, Various kinds of local connectedness, Math. Pannon. 13 (2002), no. 1, 103-116.
Janusz J. Charatonik and Alejandro Illanes, Local connectedness in hyperspaces, Rocky Mountain J. Math. 36 (2006), no. 3, 811-856.
Włodzimierz J. Charatonik, A homogeneous continuum without the property of Kelley, Topology Appl. 96 (1999), no. 3, 209-216.
Leobardo Fernández and Isabel Puga, On semi-Kelley continua, Houston J. Math. 45 (2019), no. 1, 307-315.
Jack T. Goodykoontz, Jr., More on connectedness im kleinen and local connectedness in $C(X)$, Proc. Amer. Math. Soc. 65 (1977), no. 2, 357-364.
Alejandro Illanes, Hereditarily indecomposable Hausdorff continua have unique hyperspaces $2^X$ and $C_n(X)$, Publ. Inst. Math. (Beograd) (N.S.) 89(103) (2011), 49-56.
Alejandro Illanes and Sam B. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances. Monographs and Textbooks in Pure and Applied Mathematics, 216. New York: Marcel Dekker, Inc., 1999.
J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36.
K. Kuratowski, Topology. Vol. II. New edition, revised and augmented. Translated from the French by A. Kirkor. New York-London: Academic Press and Warsaw: PWN, 1968.
Sergio Macías, On Jones' set function T and the property of Kelley for Hausdorff continua, Topology Appl. 226 (2017), 51-65.
Sergio Macías, Hausdorff continua and the uniform property of Effros, Topology Appl. 230 (2017), 338-352.
Władysław Makuchowski, On local connectedness in hyperspaces, Bull. Polish Acad. Sci. Math. 47 (1999), no. 2, 119-126.
Władysław Makuchowski, On local connectedness at a subcontinuum and smoothness of continua, Houston J. Math. 29 (2003), no. 3, 711-716.
Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182.
A. K. Misra, A note on arcs in hyperspaces Acta Math. Hungar. 45 (1985), no. 3-4, 285-288.
S. Mrówka, On the convergence of nets of sets, Fund. Math. 45 (1958), 237-246.
Sam B. Nadler, Jr., Hyperspaces of Sets. A Text with Research Questions. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49. New York-Basel: Marcel Dekker, Inc., 1978. Reprinted in: Aportaciones Matemáticas de la Sociedad Matemática Mexicana, Serie Textos #33, 2006.
Sam B. Nadler, Jr. Continuum Theory. An Introduction. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 158. New York: Marcel Dekker, Inc., 1992.
Roger W. Wardle, On a property of J. L. Kelley, Houston J. Math. 3 (1977), no. 2, 291-299.
