Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Nonhomogeneity of remainders


Authors: A. V. Arhangel’skii and J. van Mill
Journal: Proc. Amer. Math. Soc. 144 (2016), 4065-4073
MSC (2010): Primary 54D35, 54D40, 54A25
DOI: https://doi.org/10.1090/proc/13172
Published electronically: April 27, 2016
MathSciNet review: 3513561
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] A. Arhangel'skii, An addition theorem for the weight of sets lying in bicompacts, Dokl. Akad. Nauk SSSR 126 (1959), 239-241 (Russian). MR 0106444
  • [2] A. Arhangel'skii, 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
  • [3] A. Arhangel'skii, 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
  • [4] A. Arhangel'skii, Two types of remainders of topological groups, Comment. Math. Univ. Carolin. 49 (2008), no. 1, 119-126. MR 2433629
  • [5] A. Arhangel'skii, A study of remainders of topological groups, Fund. Math. 203 (2009), no. 2, 165-178. MR 2496236, https://doi.org/10.4064/fm203-2-3
  • [6] A. 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
  • [7] A. 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
  • [8] W. W. Comfort and Kenneth A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483-496. MR 0207886
  • [9] Eric K. van Douwen, Nonhomogeneity of products of preimages and $ \pi $-weight, Proc. Amer. Math. Soc. 69 (1978), no. 1, 183-192. MR 0644652
  • [10] Eric K. van Douwen, Why certain Čech-Stone remainders are not homogeneous, Colloq. Math. 41 (1979), no. 1, 45-52. MR 550626
  • [11] Eric K. van Douwen and Teodor C. Przymusiński, First countable and countable spaces all compactifications of which contain $ \beta {\bf N}$, Fund. Math. 102 (1979), no. 3, 229-234. MR 532957
  • [12] B. Efimov, On dyadic spaces, Dokl. Akad. Nauk SSSR 151 (1963), 1021-1024 (Russian). MR 0152987
  • [13] 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
  • [14] Zdeněk Frolík, Non-homogeneity of $ \beta P-P$, Comment. Math. Univ. Carolinae 8 (1967), 705-709. MR 0266160
  • [15] R. Hodel, Cardinal functions. I, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1-61. MR 776620
  • [16] István Juhász, Cardinal functions in topology--ten years later, 2nd ed., Mathematical Centre Tracts, vol. 123, Mathematisch Centrum, Amsterdam, 1980. MR 576927
  • [17] A. I. Krivoručko, The power of the set of continuous functions, Dokl. Akad. Nauk SSSR 206 (1972), 1046-1048 (Russian). MR 0315652
  • [18] K. Kunen, Weak $ P$-points in $ {\bf 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
  • [19] Jan van Mill, On the character and $ \pi $-weight of homogeneous compacta, Israel J. Math. 133 (2003), 321-338. MR 1968433, https://doi.org/10.1007/BF02773072
  • [20] R. S. Pierce, A note on complete Boolean algebras, Proc. Amer. Math. Soc. 9 (1958), 892-896. MR 0102487
  • [21] V. Popov, A perfect map need not preserve a $ G_{\delta }$-diagonal, General Topology and Appl. 7 (1977), no. 1, 31-33. MR 0431093
  • [22] Walter Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956), 409-419. MR 0080902
  • [23] Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
  • [24] 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, https://doi.org/10.1016/0166-8641(88)90006-5
  • [25] Boban Veličković, $ {\rm OCA}$ and automorphisms of $ {\mathcal {P}}(\omega )/{\rm fin}$, Topology Appl. 49 (1993), no. 1, 1-13. MR 1202874, https://doi.org/10.1016/0166-8641(93)90127-Y

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 54D35, 54D40, 54A25

Retrieve articles in all journals with MSC (2010): 54D35, 54D40, 54A25


Additional Information

A. V. Arhangel’skii
Affiliation: Moscow State University and Moscow State Pedagogical University, Moscow, Russia
Email: arhangel.alex@gmail.com

J. van Mill
Affiliation: KdV Institute for Mathematics, University of Amsterdam, Science Park 904, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
Email: j.vanMill@uva.nl

DOI: https://doi.org/10.1090/proc/13172
Keywords: Remainder, compactification, topological group, homogeneous space, first-countable, Continuum Hypothesis, Martin's Axiom
Received by editor(s): May 15, 2015
Received by editor(s) in revised form: May 19, 2015, and September 21, 2015
Published electronically: April 27, 2016
Additional Notes: The work of the first-named author is supported by RFBR, project 15-01-05369
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society