Nonhomogeneity of remainders

Arhangel’skii, A.; van Mill, J.

doi:10.1090/proc/13172

Nonhomogeneity of remainders
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by A. V. Arhangel’skii and J. van Mill PDF

Proc. Amer. Math. Soc. 144 (2016), 4065-4073 Request permission

Abstract:

We present a cardinal inequality for the number of homeomorphisms of the remainders of compactifications of nowhere locally compact spaces. As a consequence, we obtain that if $X$ is countable and dense in itself, then the remainder of any compactification of $X$ has at most continuum many homeomorphisms.

References

A. Arhangel′skiĭ, An addition theorem for the weight of sets lying in bicompacts, Dokl. Akad. Nauk SSSR 126 (1959), 239–241 (Russian). MR 0106444
A. Archangelski, Concerning the weight of topological spaces, General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961) Academic Press, New York; Publ. House Czech. Acad. Sci., Prague, 1962, pp. 72–74. MR 0174030
A. V. Arkhangel′skiĭ, Topological function spaces, Mathematics and its Applications (Soviet Series), vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the Russian by R. A. M. Hoksbergen. MR 1144519, DOI 10.1007/978-94-011-2598-7
A. V. Arhangel′skii, Two types of remainders of topological groups, Comment. Math. Univ. Carolin. 49 (2008), no. 1, 119–126. MR 2433629
A. V. Arhangel′skii, A study of remainders of topological groups, Fund. Math. 203 (2009), no. 2, 165–178. MR 2496236, DOI 10.4064/fm203-2-3
Alexander Arhangel′skii, The Baire property in remainders of topological groups and other results, Comment. Math. Univ. Carolin. 50 (2009), no. 2, 273–279. MR 2537836
Alexander Arhangel′skii and Mikhail Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, vol. 1, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. MR 2433295, DOI 10.2991/978-94-91216-35-0
W. W. Comfort and Kenneth A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483–496. MR 207886
Eric K. van Douwen, Nonhomogeneity of products of preimages and $\pi$-weight, Proc. Amer. Math. Soc. 69 (1978), no. 1, 183–192. MR 644652, DOI 10.1090/S0002-9939-1978-0644652-8
Eric K. van Douwen, Why certain Čech-Stone remainders are not homogeneous, Colloq. Math. 41 (1979), no. 1, 45–52. MR 550626, DOI 10.4064/cm-41-1-45-52
Eric K. van Douwen and Teodor C. Przymusiński, First countable and countable spaces all compactifications of which contain $\beta \textbf {N}$, Fund. Math. 102 (1979), no. 3, 229–234. MR 532957, DOI 10.4064/fm-102-3-229-234
B. Efimov, On dyadic spaces, Dokl. Akad. Nauk SSSR 151 (1963), 1021–1024 (Russian). MR 0152987
B. A. Efimov, Extremally disconnected bicompacta and absolutes (on the occasion of the one hundredth anniversary of the birth of Felix Hausdorff), Trudy Moskov. Mat. Obšč. 23 (1970), 235–276 (Russian). MR 0418016
Zdeněk Frolík, Non-homogeneity of $\beta P-P$, Comment. Math. Univ. Carolinae 8 (1967), 705–709. MR 266160
R. Hodel, Cardinal functions. I, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1–61. MR 776620
István Juhász, Cardinal functions in topology—ten years later, 2nd ed., Mathematical Centre Tracts, vol. 123, Mathematisch Centrum, Amsterdam, 1980. MR 576927
A. I. Krivoručko, The power of the set of continuous functions, Dokl. Akad. Nauk SSSR 206 (1972), 1046–1048 (Russian). MR 0315652
K. Kunen, Weak $P$-points in $\textbf {N}^{\ast }$, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 741–749. MR 588822
Jan van Mill, On the character and $\pi$-weight of homogeneous compacta, Israel J. Math. 133 (2003), 321–338. MR 1968433, DOI 10.1007/BF02773072
R. S. Pierce, A note on complete Boolean algebras, Proc. Amer. Math. Soc. 9 (1958), 892–896. MR 102487, DOI 10.1090/S0002-9939-1958-0102487-6
V. Popov, A perfect map need not preserve a $G_{\delta }$-diagonal, General Topology and Appl. 7 (1977), no. 1, 31–33. MR 431093
Walter Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956), 409–419. MR 80902
Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
J. Steprāns and H. X. Zhou, Some results on CDH spaces. I, Topology Appl. 28 (1988), no. 2, 147–154. Special issue on set-theoretic topology. MR 932979, DOI 10.1016/0166-8641(88)90006-5
Boban Veli ković, $\textrm {OCA}$ and automorphisms of ${\scr P}(\omega )/\textrm {fin}$, Topology Appl. 49 (1993), no. 1, 1–13. MR 1202874, DOI 10.1016/0166-8641(93)90127-Y