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The Topology of Free Topological Groups

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Abstract

The known explicit descriptions of the topology of free and free Abelian topological groups are collected. Examples of the application of such descriptions by various authors to study the properties of topological groups and some related topological-algebraic objects are given. Various approaches to describing the topology of free topological groups are presented and a general method for topologizing free groups is suggested.The fundamental properties of free topological groups and their structure are considered.The most important results on free topological groups obtained by various authors through the years are mentioned.

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REFERENCES

  1. P. S. Aleksandrov and B. A. Pasynkov, An Introduction to Dimension Theory. An Introduction to the Theory of Topological Spaces and General Dimension Theory [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  2. A. V. Arkhangel'skii, “On mappings related to topological groups,” Dokl. Akad. Nauk SSSR, 181, No.6, 1303–1306 (1968).

    Google Scholar 

  3. A. V. Arkhangel'skii, Topological Spaces and Continuous Mappings. Remarks on Topological Groups [in Russian], Izd. Mosk. Univ., Moscow (1969).

    Google Scholar 

  4. A. V. Arkhangel'skii, “On relations between invariants of topological groups and their subspaces,” Usp. Mat. Nauk, 35, No.3, 3–22 (1980).

    Google Scholar 

  5. A. V. Arkhangel'skii, “Classes of topological groups,” Usp. Mat. Nauk, 36, No.3, 127–146 (1981).

    Google Scholar 

  6. A. V. Arkhangel'skii, “Any topological group is a quotient group of a zero-dimensional topological group, ” Dokl. Akad. Nauk SSSR, 258, No.5, 1037–1040 (1981).

    Google Scholar 

  7. A. V. Arkhangel'skii, “Algebraic objects generated by a topological structure,” in: Itogi Nauki i Tekhniki. Algebra. Topologiya. Geometriya [in Russian], 25, Moscow (1987), pp. 141–198.

    Google Scholar 

  8. A. V. Arhangel'skii, “Topological invariants in algebraic environment,” in: Recent Progress in General Topology 2, North-Holland, Amsterdam (2002), pp. 1–57.

    Google Scholar 

  9. A. V. Arhangel'skii, O. G. Okunev, and V. G. Pestov, “Free topological groups over metrizable spaces, ” Topol. and Its Appl., 33, No.1, 63–76 (1989).

    Article  Google Scholar 

  10. V. K. Bel'nov, “On the dimension of free topological groups,” in: IVth Tiraspol Symposium on General Topology and Its Applications [in Russian], Shtiintsa, Kishinev (1979), pp. 14–15.

    Google Scholar 

  11. C. R. Borges, “Free topological groups,” J. Austral. Math. Soc., 23 (Ser. A), 360–365 (1977).

    Google Scholar 

  12. A. A. Borubaev and A. A. Chekeev, “On τ-completeness of topological groups,” Zap. Nauchn. Sem. St. Peterb. Otd. Mat. Inst. im. V. A. Steklova, Ross. Akad. Nauk (POMI), 208, Issled. Topol. 7, 103–114, 220–221 (1997).

    Google Scholar 

  13. M. M. Choban, “On some questions of the theory of topological groups,” in: General Algebra and Discrete Geometry. Mathematical Sciences [in Russian], Shtiintsa, Kishinev (1980), pp. 120–135.

    Google Scholar 

  14. M. M. Choban, “On the theory of topological algebraic systems,” Tr. Mosk. Mat. Ob-va, 48, 106–149 (1985).

    Google Scholar 

  15. M. M. Choban, “General conditions for the existence of free objects,” Acta Comment. Univ. Tartuensis, 836, 157–171 (1989).

    Google Scholar 

  16. M. M. Choban, “Some topics in topological algebra,” Topol. Appl., 54, No.1–3, 183–202 (1993).

    Article  Google Scholar 

  17. M. M. Choban, “Algebraical equivalences of topological spaces,” Matematica, No. 1(35), 12–36 (2001).

    Google Scholar 

  18. W. W. Comfort, “Problems on topological groups and other homogeneous spaces,” in: J. van Mill and G. M. Reed, Eds., Open Problems in Topology, North-Holland, Amsterdam (1990), pp. 313–347.

    Google Scholar 

  19. W. W. Comfort, K.-H. Hoffman, and D. Remus, “Topological groups and semigroups,” in: Recent Progress in General Topology, North-Holland, Amsterdam (1992), pp. 57–144.

    Google Scholar 

  20. S. S. Dumitrashku and M. M. Choban, “On free topological algebras with continuous signature,” in: Algebraic and Topological Systems [in Russian], Shtiintsa, Kishinev (1982), pp. 27–53.

    Google Scholar 

  21. K. Eda, H. Ohta, and K. Yamada, “Prime subspaces in free topological groups,” Topol. and Its Appl., 62, 163–171 (1995).

    Article  Google Scholar 

  22. R. Engelking, General Topology, Heldermann, Berlin (1989).

    Google Scholar 

  23. E. E. Enochs, “Homotopy groups of compact Abelian groups,” Proc. Amer. Math. Soc., 15, No.6, 878–881 (1964).

    Google Scholar 

  24. T. Fay, E. Ordman, and B. V. S. Thomas, “The free topological group over rationals,” Gen. Topol. Appl., 10, No.1, 33–47 (1979).

    Article  Google Scholar 

  25. J. Galindo and S. Hernandez, “Pontryagin-van Kampen reflexivity for free Abelian topological groups, ” Forum Math., 11, 399–415 (1999).

    Google Scholar 

  26. P. M. Gartside, E. A. Reznichenko, and O. V. Sipacheva, “Maltsev and Retral spaces,” Topol. and Its Appl., 80, 115–129 (1997).

    Article  Google Scholar 

  27. M. I. Graev, “Free topological groups,” Izv. Akad. Nauk SSSR. Ser. Mat., 12, 279–324 (1948).

    Google Scholar 

  28. M. I. Graev, “The theory of topological groups. I,” Usp. Mat. Nauk, 5, No.2, 3–56 (1950).

    Google Scholar 

  29. I. I. Guran, On Embeddings of Topological Groups [in Russian], Available from VINITI, No. 1483-81, Moscow (1981).

  30. I. I. Guran, “Topology on an infinite symmetric group and condensations,” Comment. Math. Univ. Carolin., 22, No.2, 311–316 (1981).

    Google Scholar 

  31. E. Hewitt and K. Ross, Abstract Harmonic Analysis. Vol. I: Structure of Topological Groups. Integration Theory, Group Representations, Academic Press, New York; Springer, Berlin (1963).

    Google Scholar 

  32. K. H. Hofmann, “An essay on free compact groups,” Lect. Notes Math., 915, 171–197 (1982).

    Google Scholar 

  33. K. H. Hofmann and S. A. Morris, “Free compact groups. I: Free compact Abelian groups,” Topol. and Its Appl., 23, No.1, 41–64 (1986).

    Article  Google Scholar 

  34. D. C. Hunt and S. A. Morris, “Free subgroups of free topological groups,” in: M. F. Newman, ed., Proc. Second Int. Conf. on the Theory of Groups, Lecture Notes in Mathematics, 372, Springer, Berlin (1974), pp. 377–387.

    Google Scholar 

  35. C. Joiner, “Free topological groups and dimension,” Trans. Amer. Math. Soc., 220, 401–418 (1976).

    Google Scholar 

  36. S. Kakutani, “Free topological groups and infinite direct product of topological groups,” Proc. Imp. Acad. Tokyo, 20, 595–598 (1944).

    Google Scholar 

  37. G. I. Kats, “An isomorphic mapping of topological groups to a direct product of first countable groups, ” Usp. Mat. Nauk, 8, No.6, 107–113 (1953).

    Google Scholar 

  38. L. L. Kiriyak and M. M. Choban, “An application of uniform structures to a study of free topological algebras,” Sib. Mat. Zh., 33, No.5, 891–904 (1992).

    Article  Google Scholar 

  39. A. Leiderman, S. A. Morris, and V. Pestov, “The free Abelian topological group and the free locally convex space on the unit interval,” J. London Math. Soc., 56, 529–538 (1997).

    Article  Google Scholar 

  40. J. Mack, S. A. Morris, and E. T. Ordman, “Free topological groups and the projective dimension of locally compact Abelian groups,” Proc. Amer. Math. Soc., 40, 303–308 (1973).

    Google Scholar 

  41. A. I. Mal'tsev, “On the general theory of algebraic systems,” Mat. Sb., 35, 3–20 (1954).

    Google Scholar 

  42. A. I. Mal'tsev, “Free topological algebras,” Izv. Akad. Nauk SSSR, Ser. Mat., 21, 171–198 (1957).

    Google Scholar 

  43. A. A. Markov, “On free topological groups,” Dokl. Akad. Nauk SSSR, 31, No.4, 299–301 (1941).

    Google Scholar 

  44. A. A. Markov, “On free topological groups,” Izv. Akad. Nauk SSSR, Ser. Mat., 9, No.1, 3–64 (1945).

    Google Scholar 

  45. E. Michael, “Local compactness and Cartesian product of quotient maps and k-spaces,” Ann. Inst. Fourier, 18, 281–286 (1968).

    Google Scholar 

  46. D. Montgomery and L. Zippin, Topological Transformation Groups, New York (1955).

  47. S. A. Morris, “Varieties of topological groups,” Bull. Austral. Math. Soc., 1, No.2, 145–160 (1969).

    Google Scholar 

  48. S. A. Morris, “Varieties of topological groups. II,” Bull. Austral. Math. Soc., 2, No.1, 1–13 (1970).

    Google Scholar 

  49. S. A. Morris, “Varieties of topological groups. III,” Bull. Austral. Math. Soc., 2, No.2, 165–178 (1970).

    Google Scholar 

  50. S. A. Morris, “Free compact Abelian groups,” Mat. Cas., 22, No.2, 141–147 (1972).

    Google Scholar 

  51. S. A. Morris, “Varieties of topological groups: A survey,” Colloq. Math., 46, No.2, 147–165 (1982).

    Google Scholar 

  52. S. A. Morris and H. B. Thompson, “Invariant metrics on free topological groups,” Bull. Austral. Math. Soc., 9, No.1, 83–88 (1973).

    Google Scholar 

  53. S. A. Morris and H. B. Thompson, “Free topological groups with no small subgroups,” Proc. Amer. Math. Soc., 46, No.3, 431–437 (1974).

    Google Scholar 

  54. T. Nakayama, “Note on free topological groups,” Proc. Imp. Acad. Tokyo, 19, 471–475 (1943).

    Google Scholar 

  55. E. C. Nummela, “Uniform free topological groups and Samuel compactifications,” Topol. and Its Appl., 13, No.1, 77–83 (1982).

    Article  Google Scholar 

  56. P. J. Nyikos, “Subsets of ωω and the Frechet-Urysohn and αi-properties, ” Topol. and Its Appl., 48, 91–116 (1992).

    Article  Google Scholar 

  57. O. G. Okunev, “A method for constructing examples of M-equivalent spaces,” Topol. and Its Appl., 36, No.2, 157–171 (1990); correction: “A method for constructing examples of M-equivalent spaces,” Topol. Appl., 49, No. 2, 191–192 (1993).

    Article  Google Scholar 

  58. O. Okunev and K. Tamano, “Lindelof powers and products of function spaces,” Proc. Amer. Math. Soc., 124, No.9, 2905–2916 (1996).

    Article  Google Scholar 

  59. V. G. Pestov, On the Structure and Embeddings of Topological Groups, Available from VINITI, No. 1495-81, Tomsk (1981).

  60. V. G. Pestov, “Some properties of free topological groups,” Vestn. Mosk. Univ. Ser. 1. Mat., Mekh., No. 1, 35–37 (1982).

  61. V. G. Pestov, “Neighborhoods of identity in free topological groups,” Vestn. Mosk. Univ. Ser. 1. Mat., Mekh., No. 3, 8–10 (1985).

  62. V. G. Pestov, “Free topological Abelian groups and Pontryagin duality,” Vestn. Mosk. Univ. Ser. 1. Mat., Mekh., No. 1, 3–5 (1986).

  63. V. G. Pestov, “Universal arrows to forgetful functors from categories of topological algebra,” Bull. Austral. Math. Soc., 48, 209–249 (1993).

    Google Scholar 

  64. V. Pestov, “Topological groups: Where to from here?” Topol. Proc., 24 421–502 (1999).

    Google Scholar 

  65. V. Pestov and K. Yamada, “Free topological groups on metrizable spaces and inductive limits,” Topol. and Its Appl., 98, 291–301 (1999).

    Article  Google Scholar 

  66. H. E. Porst, “Zur Struktur freier topologischer Gruppen,” Rostock. Math. Kolloq., No. 44, 5–20 (1991).

  67. T. Przymusinski, “Collectionwise normality and extensions of continuous functions,” Fund. Math., 98, No.1, 75–81 (1978).

    Google Scholar 

  68. S. T. Rachev and L. Ruschendorf, Mass Transportation Problems. Vol. I: Theory. Vol. II: Applications, Springer, New York-Berlin-Heidelberg (1998).

    Google Scholar 

  69. E. A. Reznichenko and O. V. Sipacheva, “Quotient mappings to words of bounded length in free topological groups,” in: General Topology. Mappings, Products, and Dimension of Spaces [in Russian], Izd. Mosk. Univ., Moscow (1995), pp. 98–119.

    Google Scholar 

  70. E. A. Reznichenko and V. V. Uspenskii, “Pseudocompact Mal'tsev spaces,” Topol. and Its Appl., 86, 83–104 (1998).

    Article  Google Scholar 

  71. M. E. Rudin, Lectures on Set Theoretic Topology, Amer. Math. Soc., Providence, Rhode Island (1975).

    Google Scholar 

  72. P. Samuel, “On universal mappings and free topological groups,” Bull. Amer. Math. Soc., 54, No.6, 591–598 (1948).

    Google Scholar 

  73. D. B. Shakhmatov, “Precalibers of σ-compact topological groups,” Mat. Zametki, 39, No.6, 859–868 (1986).

    Google Scholar 

  74. D. B. Shakhmatov, “Zerodimensionality of free topological groups and topological groups with noncoinciding dimensions,” Bull. Pol. Acad. Sci. Math., 37, No.7–12, 497–506 (1989).

    Google Scholar 

  75. D. B. Shakhmatov, “A survey of current research and open problems in the dimension theory of topological groups,” Q&A in General Topology, 8, 101–128 (1990).

    Google Scholar 

  76. D. B. Shakhmatov, “Imbeddings into topological groups preserving dimensions,” Topol. and Its Appl., 36 181–204 (1990).

    Article  Google Scholar 

  77. O. V. Sipacheva, “Topologies on free groups,” in: Vth Tiraspol Symposium on General Topology and Its Applications [in Russian], Shtiintsa, Kishinev (1985), pp. 220–222.

    Google Scholar 

  78. O. V. Sipacheva, “A description of the free group topology without application of universal uniform structures,” in: General Topology. Mappings of Topological Spaces [in Russian], Izd. Mosk. Univ., Moscow (1986), pp. 122–130.

    Google Scholar 

  79. O. V. Sipacheva, “Compact spaces with continuous Mal'tsev operation and retracts of topological groups, ” Vestn. Mosk. Univ. Ser. 1. Mat., Mekh., No. 1, 33–36 (1991).

  80. O. V. Sipacheva, “On free topological groups with the inductive limit topologies,” in: Annals of the New York Acad. Sci., 788, The New York Acad. Sci., New York (1996), pp. 188–196.

    Google Scholar 

  81. O. V. Sipacheva, “Free topological groups of spaces and their subspaces,” Topol. and Its Appl., 101, 181–212 (2000).

    Article  Google Scholar 

  82. O. V. Sipacheva, “τ-Locally invariant groups,” Topol. and Its Appl., 107, 169–182 (2000).

    Article  Google Scholar 

  83. O. V. Sipacheva, “A class of free locally convex spaces,” Mat. Sb., 194, No.3, 25–52 (2003).

    Google Scholar 

  84. O. V. Sipacheva and M. G. Tkachenko, “Thin and bounded subsets of free topological groups,” Topol. and Its Appl., 36, 143–156 (1990).

    Article  Google Scholar 

  85. O. V. Sipacheva and V. V. Uspenskii, “Free topological groups without small subgroups,” Vestn. Mosk. Univ. Ser. 1. Mat., Mekh., No. 4, 21–24 (1987).

  86. S. Swierczkowski, “Topologies in free algebras,” Proc. London Math. Soc., 14, No.55, 566–576 (1964).

    Google Scholar 

  87. H. B. Thompson, A remark on free topological groups with no small subgroups,” J. Austral. Math. Soc., 18, No.4, 482–484 (1974).

    Google Scholar 

  88. M. G. Tkachenko, “On the topology of Markov free groups,” in: Vth Symposium on General Topology, Prague (1982).

  89. M. G. Tkachenko, “On zero-dimensional topological groups,” in: Proc. of Leningrad Int. Conf. on Topology and Its Applications [in Russian], Leningrad, Nauka (1982), pp. 113–118.

    Google Scholar 

  90. M. G. Tkachenko, “On the completeness of free Abelian topological groups,” Dokl. Akad. Nauk SSSR, 269, 299–303 (1983).

    Google Scholar 

  91. M. G. Tkachenko, “On the Souslin property in free topological groups over compact spaces,” Mat. Zametki, 34, No.4, 601–607 (1983).

    Google Scholar 

  92. M. G. Tkachenko, “On the completeness of topological groups,” Sib. Mat. Zh., 25, No. 1, 146–158 (1984).

    Google Scholar 

  93. M. G. Tkachenko, “On the topology of free groups over compact spaces,” in: Mapping and Functors [in Russian], Izd. Mosk. Univ., Moscow (1984), pp. 122–137.

    Google Scholar 

  94. M. G. Tkachenko, “On a spectral decomposition of free topological groups,” Usp. Mat. Nauk, 39, No.2 (236), 191–192 (1984).

    Google Scholar 

  95. M. G. Tkacenko, “On topologies of free groups,” Czechoslovak Math. J., 34, No.4, 541–551 (1984).

    Google Scholar 

  96. M. G. Tkachenko, “Zero-dimensionality of free topological groups,” C. R. Acad. Bulgare Sci., 38, No.2, 173–174 (1985).

    Google Scholar 

  97. M. G. Tkachenko, “On some properties of free topological groups,” Mat. Zametki, 37, No.1 (1985).

    Google Scholar 

  98. M. G. Tkachenko, “Strict collectionwise normality and countable compactness in free topological groups, ” Sib. Mat. Zh., 28, No.5, 167–177 (1987).

    Google Scholar 

  99. M. G. Tkacenko, “Free topological groups and inductive limits,” Topol. and Its Appl., 60, 1–12 (1994).

    Article  Google Scholar 

  100. M. G. Tkacenko, “Topological groups for topologists. I, II,” Bol. Soc. Mat. Mexicana, 5, 735–753 (1999); 6, 1–45 (2000).

    Google Scholar 

  101. V. V. Uspenskii, “A topological group generated by a Lindelof Σ-space has the Souslin property, ” Dokl. Akad. Nauk SSSR, 265, No.4, 823–826 (1982).

    Google Scholar 

  102. V. V. Uspenskii, “On subgroups of free topological groups,” Dokl. Akad. Nauk SSSR, 285, No.5, 1070–1072 (1985).

    Google Scholar 

  103. V. V. Uspenskii, “On continuous images of Lindelof topological groups,” Dokl. Akad. Nauk SSSR, 285, No.4, 824–827 (1985).

    Google Scholar 

  104. V. V. Uspenskii, “Topological group and Dugundji compacta,” Mat. Sb., No. 8, 1092–1118 (1989).

  105. V. V. Uspenskii, “The Mal'tsev operation on countably compact spaces,” Comment. Math. Univ. Carolin., 30, No.2, 395–402 (1989).

    Google Scholar 

  106. V. V. Uspenskii, “Free topological groups of metrizable spaces,” Izv. Akad. Nauk SSSR. Ser. Mat., 54, No.6, 1295–1319 (1990).

    Google Scholar 

  107. S. W. Williams, “Box products,” in: K. Kunen and J. E. Vaughan, Eds., Handbook of Set-Theoretic Topology, North-Holland, Amsterdam-New York-Oxford-Tokyo (1984), pp. 169–200.

    Google Scholar 

  108. K. Yamada, “Characterizations of a metrizable space X such that every A n (X) isa k-space,” Topol. and Its Appl., 49, 75–94 (1993).

    Article  Google Scholar 

  109. K. Yamada, “Tightness of free Abelian topological groups and of finite products of sequential fans, ” Topol. Proc., 22, 363–381 (1997).

    Google Scholar 

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  1. Department of General Topology and Geometry, Mechanics and Mathematics Faculty, Moscow State University, Leninskie gory, Moscow, 119992, Russia

    O. V. Sipacheva

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 2, pp. 99–204, 2003.

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Sipacheva, O.V. The Topology of Free Topological Groups. J Math Sci 131, 5765–5838 (2005). https://doi.org/10.1007/s10958-005-0445-z

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