Indecomposable continua in Stone-Čech compactifications

Bellamy, David P.; Rubin, Leonard R.

doi:10.1090/S0002-9939-1973-0315670-X

Indecomposable continua in Stone-Čech compactifications

Authors: David P. Bellamy and Leonard R. Rubin
Journal: Proc. Amer. Math. Soc. 39 (1973), 427-432
MSC: Primary 54D35; Secondary 54F20
DOI: https://doi.org/10.1090/S0002-9939-1973-0315670-X
MathSciNet review: 0315670
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Abstract: We show that if is a continuum irreducible from to , which is connected im Kleinen and first countable at , and if $X = Y - \{ b\}$ , then $\beta X - X$ is an indecomposable continuum. Examples are given showing that both first countability and connectedness im Kleinen are needed here. We also show that $\beta [0,1) - [0,1)$ has a strong near-homogeneity property.

[1] David P. Bellamy, Continua for which the set function 𝑇 is continuous, Trans. Amer. Math. Soc. 151 (1970), 581–587. MR 271910, https://doi.org/10.1090/S0002-9947-1970-0271910-9
[2] David P. Bellamy, A non-metric indecomposable continuum, Duke Math. J. 38 (1971), 15–20. MR 271911
[3] -, Topological properties of compactifications of a half-open interval, Ph.D. Thesis, Michigan State University, East Lansing, Mich., 1968.
[4] H. S. Davis, A note on connectedness im kleinen, Proc. Amer. Math. Soc. 19 (1968), 1237–1241. MR 254814, https://doi.org/10.1090/S0002-9939-1968-0254814-3
[5] H. S. Davis, D. P. Stadtlander, and P. M. Swingle, Properties of the set functions 𝑇ⁿ, Portugal. Math. 21 (1962), 113–133. MR 142108
[6] H. S. Davis, D. P. Stadtlander, and P. M. Swingle, Semigroups, continua and the set functions 𝑇ⁿ, Duke Math. J. 29 (1962), 265–280. MR 146805
[7] R. F. Dickman Jr., A necessary and sufficient condition for 𝛽𝑋\𝑋 to be an indecomposable continuum, Proc. Amer. Math. Soc. 33 (1972), 191–194. MR 295296, https://doi.org/10.1090/S0002-9939-1972-0295296-6
[8] R. F. Dickman, L. R. Rubin, and P. M. Swingle, Characterization of 𝑛-spheres by an excluded middle membrane principle, Michigan Math. J. 11 (1964), 53–59. MR 161315
[9] R. F. Dickman, L. R. Rubin, and P. M. Swingle, Irreducible continua and generalization of hereditarily unicoherent continua by means of membranes, J. Austral. Math. Soc. 5 (1965), 416–426. MR 0188997
[10] James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
[11] R. W. FitzGerald and P. M. Swingle, Core decomposition of continua, Fund. Math. 61 (1967), 33–50. MR 224063, https://doi.org/10.4064/fm-61-1-33-50
[12] Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
[13] John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
[14] Robert P. Hunter, On the semigroup structure of continua, Trans. Amer. Math. Soc. 93 (1959), 356–368. MR 109194, https://doi.org/10.1090/S0002-9947-1959-0109194-X
[15] K. Kuratowski, Topology. Vol. II, New edition, revised and augmented. Translated from the French by A. Kirkor, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. MR 0259835