Compact condensations of Hausdorff spaces


We continue to study one of the classic problems in general topology raised by P. S. Alexandrov: when a Hausdorff space X has a continuous bijection (a condensation) onto a compactum? We concentrate on the situation when not only X but also \(X\setminus Y\) can be condensed onto a compactum whenever the cardinality of Y does not exceed certain \(\tau\).

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  1. 1.

    P. S. Aleksandrov, On some basic directions in general topology, Uspekhi Mat. Nauk, 19 (120) (1964), 3–46 (in Russian); translation in Russian Math. Surveys, 19 (1964), 1–39; Erratum: Corrections to the article ``On some basic directions in general topology", Uspekhi Mat. Nauk, 20 (121) (1965), 253–254

  2. 2.

    P. S. Alexandrov and H. Hopf, Topologie. I (Berlin, 1935)

  3. 3.

    Arhangel'skii, A.V.: On condensations of \(C_p\)-spaces onto compacta. Proc. Amer. Math. Soc. 128, 1881–1883 (2000)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Arhangel'skii, A.V., Pavlov, O.: A note on condensations of \(C_p(X)\) onto compacta. Comment. Math. Univ. Carol. 43, 485–492 (2002)

    MATH  Google Scholar 

  5. 5.

    A. V. Arhangel'skii and V. I. Ponomarev, Basics of general topology in problems and exercises, M. Science (1974) (in Russian); Fundamentals of General Topology. Problems and Exercises, Mathematics and Its Applications, D. Reidel Publishing Co. (Dordrecht, 1984)

  6. 6.

    Banach, S.: Problem 26. Colloq. Math. 1, 150 (1948)

    Google Scholar 

  7. 7.

    Bell, M.G.: Generalized dyadic spaces. Fund. Math. 125, 47–58 (1985)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bel'nov, V.K.: Compressions onto compacta. Dokl. Akad. Nauk SSSR 193, 506–509 (1970). (in Russian)

    MathSciNet  Google Scholar 

  9. 9.

    Belugin, V.I.: Contractions onto bicompact. Dokl. Akad. Nauk SSSR 207, 259–261 (1972). (in Russian)

    MathSciNet  Google Scholar 

  10. 10.

    Belugin, V.I.: Contractions onto bicompact. Dokl. Akad. Nauk Bolg. 28, 1447–1449 (1975)

    MathSciNet  Google Scholar 

  11. 11.

    V. I. Belugin, A. V. Osipov and E. G. Pytkeev, On classes of subcompact spaces, Math. Notes, to appear

  12. 12.

    Comfort, W.W., Hager, A.W., van Mill, J.: Compact condensations and compactifications. Topology Appl. 259, 67–79 (2019)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Corson, H.H.: Normality in subsets of product spaces. Amer. J. Math. 81, 785–796 (1959)

    MathSciNet  Article  Google Scholar 

  14. 14.

    R. Engelking, General Topology, PWN (Warsaw, 1977); Mir (Moscow, 1986)

  15. 15.

    N. Hadzhiivanov, Extension of mappings into spheres and P. S. Aleksandrov's problem of bicompact compressions, Dokl. Akad. Nauk SSSR, 194 (1970), 525–527 (in Russian)

  16. 16.

    Iliadis, S.: Some properties of absolutes. Dokl. Akad. Nauk SSSR 152, 798–800 (1963). (in Russian)

    MathSciNet  Google Scholar 

  17. 17.

    Katetov, M.: On \(H\)-closed extensions of topological spaces. Časopis Pěst. Mat. Fys. 72, 17–32 (1947)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Kulpa, W., Turzański, M.: Bijections onto compact spaces. Acta Univ. Carolin. Math. Phys. 29, 43–49 (1988)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    R. L. Moore, Foundations of Point Set Theory, AMS Colloquium Publ., XIII, Amer. Math. Soc. (Providence, RI, 1932)

  20. 20.

    Mrówka, S.: Mazur theorem and \(m\)-adic spaces. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18, 299–305 (1970)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Osipov, A.V., Pytkeev, E.G.: On the problem of condensation onto compacta. Dokl. Akad. Nauk 488, 130–132 (2019)

    Google Scholar 

  22. 22.

    A. S. Parhomenko, Über eineindeutige stetige Abbildungen, Mat. Sb. (N.S.), 5, (1939), 197–210 (in Russian)

  23. 23.

    Parhomenko, A.S.: Über eineindeutige stetige Abbildungen auf kompakte Raume. Izv. Akad. Nauk SSSR Ser. Mat. 5, 225–232 (1941)

    MathSciNet  Google Scholar 

  24. 24.

    Porter, J.R., Woods, R.G.: Extensions and Absolutes of Hausdorff Spaces. Springer-Verlag (1988)

    Google Scholar 

  25. 25.

    V. V. Proizvolov, On one-to-one continuous mappings of topological spaces, Mat. Sb. (N.S.), 68 (1965), 417–431

  26. 26.

    Pytkeev, E.G.: Upper bounds of topologies. Math. Notes 20, 831–837 (1976)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Pytkeev, E.G.: On the theory of condensations onto compact metric spaces. Dokl. Akad. Nauk SSSR 233, 1046–1048 (1977)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    E. G. Pytkeev, On the theory of one-to-one continuous mappings. Ural. Gos. Univ. Mat. Zap., Issled. Sovremen. Mat. Anal., 10 (1977), 121–132 (in Russian)

  29. 29.

    Pytkeev, E.G.: Hereditarily plumed spaces. Math. Notes 28, 761–769 (1980)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Pytkeev, E.G.: On condensations onto compact Hausdorff spaces. Dokl. Akad. Nauk SSSR 265, 819–823 (1982)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Raukhvarger, I.L.: On condensations into compacts. Dokl. Akad. Nauk SSSR 66, 13–15 (1949)

    Google Scholar 

  32. 32.

    Reiter, H.: Spaces with compact subtopologies. Rocky Mountain J. Math. 2, 239–247 (1972)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Smirnov, Y.M.: Contractions to bicompacta and their connection with retractions and bicompact extensions. Fund. Math. 63, 199–211 (1968). (in Russian)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Y. Tanaka, Quotient spaces and decompositions, in: Encyclopedia of General Topology, K. P. Hart, Jun-iti Nagata and J. E. Vaughan, Eds., Elsevier Science (2003), pp. 43–46

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The authors would like to thank the referee for careful reading and valuable comments and suggestions. The work was performed as part of research conducted in the Ural Mathematical Center.

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Correspondence to A. V. Osipov.

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Belugin, V.I., Osipov, A.V. & Pytkeev, E.G. Compact condensations of Hausdorff spaces. Acta Math. Hungar. (2021).

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Key words and phrases

  • \(a_{\tau}\)-space
  • subcompact space
  • continuous decomposition
  • weakly dyadic compact
  • condensation

Mathematics Subject Classification

  • 54C10
  • 54D30