Skip to main content
SpringerLink
Log in

Pseudocompactness in the Realm of Topological Transformation Groups

  • Chapter
  • First Online:
Pseudocompact Topological Spaces

Part of the book series: Developments in Mathematics ((DEVM,volume 55))

Abstract

In what follows, G is a Hausdorff topological group with identity element e. A topological transformation group or a G-space with phase group G is a triple \(\langle X, G, \alpha \rangle \), where X is a topological space and \(\alpha \).

This work was supported in part by grant IN-115717 from PAPIIT (UNAM).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. N. Antonyan, Equivariant embeddings and compactifications of free \(G\)-spaces, Internat. J. Math. Math. Sci. 1 (2003), 1–14.

    Google Scholar 

  2. N. Antonyan, On the maximal \(G\)-compactification of products of two \(G\)-spaces, Int. J. Math. Sci. (2006) 1–9 (Article ID 93218).

    Google Scholar 

  3. N. Antonyan, Intrinsic characterization of \(G\)-pseudocompact spaces, Houston J. Math. 33, (2) (2007) 519–530.

    Google Scholar 

  4. N. Antonyan, Extension of \(G\)-uniform maps, Topology Appl. 230 (2017), 233–243.

    Google Scholar 

  5. N. Antonyan and S. Antonyan, Free \(G\)-spaces and maximal equivariant compactifications, Annali di Matematica, 184 (2005), 407–420.

    Article  MathSciNet  Google Scholar 

  6. S. A. Antonyan, \(G\)-pseudocompact and \(G\)-Hewitt spaces, Russian Math. Surv. 35 (6) (1980), 241–245.

    Google Scholar 

  7. S. A. Antonyan, New proof of existance of a bicompact \(G\)-extension, Comment Math. Univ. Carolin. 22 (4) (1981), 761–772.

    Google Scholar 

  8. S. A. Antonyan, Equivariant pseudocompact and Hewitt spaces, in: Proc. Internat. Topol. Conf. (Leningrad, August 1982), Leningrad Branch Steklov Math. Inst. Publ. 1983, 17–26 (in Russian).

    Google Scholar 

  9. S. A. Antonyan, Equivariant embeddings and \(\omega \)- bounded groups (Russian), Vestnik Moskovskogo Universiteta, Matematika 49 (1) (1994), 16–22; English translat.: Moscow Univ. Math. Bull. 49 (1) (1994), 13–16.

    Google Scholar 

  10. S. Antonyan, M. Sanchis, Extension of locally pseudocompact group actions, Ann. Mat. Pura Appl. 4 (3) (2002), 239–246.

    Article  MathSciNet  Google Scholar 

  11. S. A. Antonyan and Yu. M. Smirnov, Universal objects and bicompact extensions for topological transformation groups, Doklady. Akad. Nauk S.S.S.R. 257 (3) (1981), 521–526; English translat.: Soviet Math. Doklady 23 (2) (1981), 279–284.

    Google Scholar 

  12. R. Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (4) (1946), 593–610.

    Article  MathSciNet  Google Scholar 

  13. A. V. Arhangel’skii, Functional tightness, \(Q\)-spaces, and \(\tau \)-embeddings, Comment. Math. Univ. Carolin. 24 (1983), 105–120.

    Google Scholar 

  14. A. V. Arkhangel’skii, Spaces of functions in the topology of the pointwise convergence, and compacta, Russian Math. Surv. 39 (5) (1984), 11–50.

    Google Scholar 

  15. A. V. Arhangel’skii, Moscow spaces, Pestov-Tkačenko problem and \(C\)-embeddings, Comment. Math. Univ. Carol. 43 (3) (2000), 585–595.

    Google Scholar 

  16. A. V. Arkhangel’skii, Topological homogeneity, topological groups and their continuous images, Russian Math. Surveys 42 (1987), 83–131. Russian original in: Uspekhy Mat. Nauk 42 (1987), 69–105.

    Article  MathSciNet  Google Scholar 

  17. A. V. Arhangel’skii, On power homogeneous spaces, Topology Appl. 122 (2002), 15–33.

    Article  MathSciNet  Google Scholar 

  18. A. V. Arhangel’skii, M. Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, 1. Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.

    Book  Google Scholar 

  19. N Blanchard, M. Jourlin, La topologie de la convergence borne sur les algbres de fonctions continues Publ. Dép. Math. (Lyon) 6 (1969) fasc. 2, 85–96.

    Google Scholar 

  20. J. L. Blasco, M. Sanchis, On the product of two \(b_{f}\)-spaces, Acta Math. Acad. Sci. Hungar. 62 (1993), 111–118.

    Google Scholar 

  21. H. Buchwalter, Topologies et compactologies, Publ. Dép. Math. (Lyon) 6 (1969), 1–74.

    Google Scholar 

  22. H. Buchwalter, Produit topologique, produit tensoriel et c-replétion, Bull. Soc. Math. France Suppl. Mém. 31-32 (1972), 51–71 (in French).

    Article  MathSciNet  Google Scholar 

  23. G. Bredon, Introduction to compact transformation groups, Academic Press, 1972.

    Google Scholar 

  24. W. W. Comfort, A. W. Hager, Uniform continuity in topological groups, in: Symposia Mathematica, vol. XVI, Academic Press, New York, 1975, pp. 269–290.

    Google Scholar 

  25. W. W. Comfort, F. J. Trigos-Arieta, Locally pseudocompact pseudocompact groups, Topology Appl. 62 (1995), 263–280.

    Google Scholar 

  26. D. Dikranjan, D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on abelian groups, Topology Appl. 151 (1-3) (2005), 2–54.

    Article  MathSciNet  Google Scholar 

  27. E. K. van Douwen, \(\beta X\) and fixed-point free maps, Topology Appl. 51(2) (1993), 191–195.

    Google Scholar 

  28. J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966.

    Google Scholar 

  29. R. Engelking, General Topology, PWN-Pol. Sci. Publ., Warsaw, 1977.

    Google Scholar 

  30. L. R. Ford, Homeomorphism groups and coset spaces, Trans. Amer. Math. Soc. 77 (3) (1954), 490–497.

    Article  MathSciNet  Google Scholar 

  31. Z. Frolik, The topological product of two pseudocompact spaces, Czechoslovak Math. J. 10 (1960), 339–349.

    Google Scholar 

  32. L. Gillman, M. Jerison, Rings of Continuous Functions, The University Series in Higher Mathematics, D. Van Nostrand, Princeton, NJ, 1960.

    Chapter  Google Scholar 

  33. I. Glicksberg, Stone-Čech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369–382.

    MathSciNet  MATH  Google Scholar 

  34. F. González, M. Sanchis, Diedonné completion and \(b_f\)-group actions, Topology Appl. 153 (2006), 3320–3326.

    Google Scholar 

  35. L. Google and M. Megrelishvili, Semigroup actions: proximities, compactifications and normality. Topology Proc. 35 (2010) 1-35.

    Google Scholar 

  36. I. I. Guran, On topological groups close to being Lindelöf, Soviet Math. Dokl. 23 (1981), 173–175.

    Google Scholar 

  37. S. Hernández, M. Sanchis, M. Tkačenko, Bounded sets in spaces and topological groups, Topology Appl. 101 (2000), 21–43.

    Article  MathSciNet  Google Scholar 

  38. M. Hušek, Hewitt realcompactification of products, Topics in topology (Proc. Colloq., Keszthely, 1972), pp. 427–435. Colloq. Math. Soc. Janos Bolyai, vol. 8, North-Holland, Amsterdam, 1974.

    Google Scholar 

  39. T. Isiwata, Mappings and spaces, Pacific J. Math. 20 (1967), 455–480.

    Article  Google Scholar 

  40. J. L. Kelley, General Topology, D. Van Nostrand Company, Inc., Toronto-London-New York, 1957.

    Google Scholar 

  41. K. L. Kozlov, V. A. Chatyrko, Topological transformation groups and Dugundji compacta, Mat. Sb. 201 (1) (2010), 103–128.

    Article  MathSciNet  Google Scholar 

  42. K. L. Kozlov, Rectangular conditions in products and equivariant completions, Topology Appl. 159 (7) (2012), 1863–1874.

    Article  MathSciNet  Google Scholar 

  43. K. L. Kozlov, Spectral decomposition of spaces induced by spectral decomposition of acting groups, Topology Appl. 160 (11) (2013), 1188–1205.

    Article  MathSciNet  Google Scholar 

  44. M. Megrelishvili, Equivariant normality, Bull. Acad. Sci. Georgian SSR 111 (1) (1983), 17–19.

    Google Scholar 

  45. M. Megrelishvili, A Tykhonov \(G\)-space not admitting a compact Hausdorff \(G\)-extension or \(G\)-linearization, Russian Math. Surv. 43 (2) (1988), 177–178.

    Google Scholar 

  46. M. Megrelishvili and T. Scarr, Constructing Tychonoff \(G\)-spaces which are not \(G\)-Tychonoff, Topol. Appl. 86 (1998), 69–81.

    Google Scholar 

  47. N. Norman, Countably compact and pseudocompact products, Czechoslovak Math. J. 19 (94) (1969), 390–397.

    Google Scholar 

  48. H. Ohta, M. Sakai, K-I. Tamano, Perfect \(\kappa \)-normality in product spaces, in: Papers on General Topology and Applications, in: Annals of the New York Academy of Sciences, vol. 704, 1993, pp. 279–289.

    Google Scholar 

  49. R. Palais The classification of \(G\)-spaces, Memoirs of the AMS 36, 1960.

    Google Scholar 

  50. V. G. Pestov, A topological transformation group without non-trivial equivariant compactifications, Advances Math. 311 (2017) 1–17.

    Article  MathSciNet  Google Scholar 

  51. R. Pupier, Topological completion of a product, Rev. Roumaine Math. Pures Appl.19 (1974), 925–933.

    Google Scholar 

  52. E. Reznichenko, Extensions of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topology Appl. 59 (3) (1994), 233–244.

    Article  MathSciNet  Google Scholar 

  53. M. Sanchis, Continuous functions on locally pseudocompact groups, Topology Appl. 86 (1) (1998), 5–23.

    Article  MathSciNet  Google Scholar 

  54. E. V. Shchepin, On \(\kappa \)-metrizable spaces, Math. USSR Izv. 14 (1980), 407–440. Russian original in: Izv. AN SSSR Ser. Mat. 43 (1979), 442–478.

    Article  Google Scholar 

  55. A. Sokolovskaya, G-compactifications of pseudocompact G-spaces, Topology Appl., 155 (2008), 342–346.

    Article  MathSciNet  Google Scholar 

  56. M. G. Tkačenko, The notion of \(o\)-tightness and \(C\)-embedded subspaces of products, Topology Appl. 15 (1) (1983), 93–98.

    Google Scholar 

  57. M. Tkachenko, Compactness type properties in topological groups, Czech. Math. J. 38 (1988), 324–341.

    Google Scholar 

  58. C. Todd, On the compactification of products, Canad. Math. Bull. 14 (1971), 591–592.

    Article  MathSciNet  Google Scholar 

  59. V. V. Uspenskii, Topological groups and Dugundji compact spaces, Math. USSR-Sb. 67 (2) (1990), 555–580.

    Article  MathSciNet  Google Scholar 

  60. J. de Vries, Can every Tychonoff G-space equivariantly be embedded in a compact Hausdorff G-space?, Math. Centrum, Amsterdam, afd. Zuivere Wisk. 36, 1975.

    Google Scholar 

  61. J. de Vries, Topological Transformation Groups 1, Mathematical Centre Tracts 65, Math. Centrum, Amsterdam, 1975.

    Google Scholar 

  62. J. de Vries, Equivariant embeddings of \(G\)-spaces, in: General Topology and its Relations to Modern Analysis and Algebra IV, Part B, pp. 485–493, Proc. 4-th Prague Topol. Symp. (1976), Prague, 1977.

    Google Scholar 

  63. J. de Vries, On the existence of \(G\)-compactifications, Bull. Acad. Polon. Sci. Ser. Math. 26 (1978), 275–280.

    Google Scholar 

  64. J. de Vries, \(G\)-spaces: Compactifications and pseudocompactness, in: Topology and Applications, Colloquia Math. SOC. Janos Bolyai 41 (1983), 655–666.

    Google Scholar 

  65. J. de Vries, On the \(G\)-compactification of products, Pac. J. Math. 26 (2) (1984), 447–470.

    Google Scholar 

  66. J. de Vries, Elements of Topological Dynamics, Kluwer Academic Publ., Dordrecht-Boston-London, 1993.

    Google Scholar 

  67. L. Waelbroek, Topological Vector Spaces and Algebras, Lectures at the Instituto de Matematica Pura y Aplicada, Rio de Janeiro, 1970.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Escuela de Ingeniería y Ciencias, Instituto Tecnológico de Estudios Superiores de Monterrey, 14380, Ciudad de Mexico, Mexico

    N. Antonyan

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de Mexico, 04510, Ciudad de Mexico, Mexico

    S. Antonyan

  3. Institut de Matemàtiques i Aplicacions de Castelló, Universitat Jaume I de Castelló, 12071, Castelló, España, Spain

    M. Sanchis

Authors
  1. N. Antonyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Antonyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Sanchis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Antonyan .

Editor information

Editors and Affiliations

  1. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México-Campus Morelia, Morelia, Michoacán, Mexico

    Michael Hrušák

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad de México, Mexico

    Ángel Tamariz-Mascarúa

  3. Departamento de Matemáticas, Universidad Autónoma Metropolitana, Ciudad de México, Mexico

    Mikhail Tkachenko

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Antonyan, N., Antonyan, S., Sanchis, M. (2018). Pseudocompactness in the Realm of Topological Transformation Groups. In: Hrušák, M., Tamariz-Mascarúa, Á., Tkachenko, M. (eds) Pseudocompact Topological Spaces. Developments in Mathematics, vol 55. Springer, Cham. https://doi.org/10.1007/978-3-319-91680-4_7

Download citation

Publish with us

Policies and ethics

Search

Navigation