Doklady Mathematics

, Volume 100, Issue 2, pp 430–432 | Cite as

On the Problem of Condensation onto Compact Spaces

  • A. V. OsipovEmail author
  • E. G. Pytkeev


Assuming the continuum hypothesis CH, it is proved that there exists a perfectly normal compact topological space Z and a countable set \(E \subset Z\) such that \(Z{\backslash }E\) is not condensed onto a compact space. The existence of such a space answers (in CH) negatively to V.I. Ponomarev’s question as to whether every perfectly normal compact space is an \(\alpha \)-space. It is proved that, in the class of ordered compact spaces, the property of being an \(\alpha \)-space is not multiplicative.



  1. 1.
    S. Banach, Colloq. Math. 1, 150 (1948).Google Scholar
  2. 2.
    V. I. Belugin, Dokl. Bolg. Akad. Nauk 28 (11), 1447–1449 (1975).MathSciNetGoogle Scholar
  3. 3.
    I. L. Raukhvarger, Dokl. Akad. Nauk SSSR 66 (13), 13–15 (1949).Google Scholar
  4. 4.
    V. I. Ponomarev, “On problems in the theory of topological spaces,” Proceedings of the 3rd Tiraspol Symposium on General Topology and Applications (Shtiiza, Chisinau, 1973), pp. 100–102.Google Scholar
  5. 5.
    V. I. Belugin, “Condensations onto bicompact spaces of subspaces of ordered bicompact spaces,” Collected Papers on Topology and Set Theory (Izhevsk, 1982), pp. 3–8 [in Russian].Google Scholar
  6. 6.
    E. G. Pytkeev, Dokl. Akad. Nauk SSSR 265 (4), 819–823 (1982).MathSciNetGoogle Scholar
  7. 7.
    V. V. Fedorchuk, Dokl. Akad. Nauk SSSR 182 (2), 275–277 (1968).MathSciNetGoogle Scholar
  8. 8.
    E. G. Pytkeev, Math. Notes 28 (4), 761–769 (1980).CrossRefGoogle Scholar
  9. 9.
    P. S. Alexandroff and P. S. Urysohn, Memoir on Compact Topological Spaces (Nauka, Moscow, 1971) [in Russian].Google Scholar
  10. 10.
    A. V. Arkhangel’skii and V. I. Ponomarev, Fundamentals of General Topology: Problems and Exercises (Nauka, Moscow, 1974; Reidel, Dordrecht, 1984).Google Scholar
  11. 11.
    V. I. Belugin, Candidate’s Dissertation in Mathematics and Physics (Sverdlovsk, 1975).Google Scholar
  12. 12.
    E. G. Pytkeev, Dokl. Akad. Nauk SSSR 233 (6), 1046–1048 (1977).MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of SciencesYekaterinburgRussia
  2. 2.Ural Federal UniversityYekaterinburgRussia

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