$o$-Boundedness of free objects over a Tychonoff space

L.S.Zdomskyy

In this paper we characterize various sorts of boundedness of the free (abelian) topological group $F(X)$ ($A(X)$) as well as the free locally-convex linear topological space $L(X)$ in terms of properties of a Tychonoff space $X$. These properties appear to be close to so-called selection principles, which permits us to show, that (it is consistent with ZFC that) the property of Hurewicz (Menger) is $l$-invariant. This gives a method of construction of $OF$-undetermined topological groups with strong combinatorial properties.

1. Архангельский A.В. Об отображениях, связанных с топологическими группами, ДАН СССР~181 (1968), 1303--1306.

2. Архангельский A.В. О соотношениях между инвариантами топологических групп и их подпространств, Успехи Мат. Наук~35 (1980), 3--22.

3. Архангельский A.В. Топологические пространства функций. -- М.: Изд. Моск. Ун-та, 1989. -- 222~с.

4. Banakh T., On index of total boundedness of (strictly) $o$-bounded groups, Topology Appl. 120 (2002), 427--439.

5. Banakh T., Nikolas P., Sanchis M. Filter games and pathologic subgroups of $\mathbb{R}^\omega$, to appear in J. Aust. Math. Soc..

6. Banakh T., Zdomsky L. Selection Principles and Games on multicovered spaces and their applications in topological algebra, in preparation.

7. Bartoszynski T., Tsaban B. Hereditary Topological Diagonalizations and the Menger-Hurewicz Conjectures, to appear in Proc. Amer. Math. Soc., \\ http://arxiv.org/abs/math.LO/0208224

8. Bouziad A. Le degr\'e de Lindel\"of est $l$-invariant, Proc. Amer. Math. Soc. 129 (2001), 913--919.

9. Chaber J., Pol R. A remark on Fremlin--Miller theorem concerning the Menger property and Michael concentrated sets, preprint.

10. Энгелькинг Р. Общая топология. -- М.:``Мир'', 1986. -- 751~с.

11. Gerlits J., Nagy Zs. Some properties of $C(X)$, I, Topology Appl. $\mathbf{14}$ (2) (1982), 151--163.

12. Гуран И.И. Топологические группы и свойства их подпространств, Канд. дис., Москва, 1981. -- 69~с.

13. Гуран I.Й., Зарічний M.М. Елементи теорії топологічних груп. -- Київ: УМК ВО, 1991. -- 76~с.

14. Hernandes C. Topological groups close to being $\sigma$-compact, Topology Appl. 102~(2000), 101--111.

15. Hernandes C., Robbie D., Tkachenko M. Some properties of $o$-bounded and strictly $o$-bounded topological groups, Appl. Gen. Topol. 1 (2000), 29--43.

16. Just W., Miller A.~W., Scheepers M., Szeptycki P.~J. The combinatorics of open covers II\/, Topology Appl. 73 (1996), 241--266.

17. Krawczyk A., Michalewski H. An example of a topological group, Topology Appl. 127 (2003), 325--330.

18. Lelek A. Some cover properties of spaces, Fund. Math. LXIV (1969), 209--218.

19. Пестов В.Г. Некоторые свойства свободных топологических групп, Вестн. Москов. Ун-та. 1 (1982), 35--37.

20. Пестов В.Г. Некоторые топологические свойства, сохраняемые отношением $M$-екви\-ва\-лент\-нос\-ти, Успехи Мат. Наук 39 (1984), no.6, 203--304.

21. Pol R. A function space $C(X)$ which is weakly Lindel\"of but not weakly compactly generated, Studia Math. 64 (1979), 279--285.

22. Roelcke W., Dierolf S. Uniform structures on topological groups and their quotients, McGraw-Hill International Book Company, 1981.

23. Scheepers M. A direct proof of a theorem of Telgarsky, Proc. Amer. Math. Soc. 123(11) (1995), 3483--3485.

24. Scheepers M. Combinatorics of open covers I: Ramsey Theory, Topology Appl. 69~(1996), 31--62.

25. Scheepers M., Tsaban B. The combinatorics of Borel covers, Topology Appl. 121~(2002), 357--382.

26. Telgarsky R. Topological games: on the 50th anniversary of the Banach-Mazur game, Rocky Mountain J. Math., 17 (1987), no.3, 227--276.

27. Telgarsky R. On sieve-complete and compact-like spaces, Topology Appl. 16 (1983), 63--68.

28. Telgarsky R. Spaces defined by topological games, Fund. Math. CXVI (1983), 189--207.

29. Telgarsky R. On games of Tops, Math. Scand. 54 (1984), 170--176.

30. Tkachenko M. Introduction to topological groups, Topology Appl. 86 (1998), 179--231.

31. Tkachenko M. Topological groups for topologists: part II, Bol. Soc. Mat. Mexicana 3 (2000), no.3, 1--41.

32. Tkachuk V. Some non-multiplicative properties are $l$-invariant, Comment. Math. Univ. Carolin. 38 (1997), 169--175.

33. Tsaban B. $o$-Bounded groups and other topological groups with strong combinatorial properties, Proc. Amer. Math. Soc., to appear. \\ http://arxiv.org/abs/math.GN/0307225

34. Tsaban B. Selection principles in mathematics: A milestone of open problems, Note Mat. 22 (2003/2004), no. 22, 179--208. \\ http://arxiv.org/abs/math.GN/0312182

35. Tsaban B., Zdomskyy L. Scales, Fields, and a problem of Hurewicz, submitted to J.~Amer. Math. Soc.. http://arxiv.org/abs/math.GN/0507043.

36. Успенский В.В. О топологии свободного локально-выпуклого пространства, ДАН СССР 270 (1983), 1334--1337.

37. Vaughan J. Small uncountable cardinals and topology, in: Open problems in topology (eds. J.~van Mill, G.M.~Reed), Elsevier Sci. Publ., 1990, 197--216.

38. Velichko N.~V. The Lindelof property is $l$-invariant, Topology Appl. 89 (1998), 277--283.

39. Zdomskyy L. A semifilter approach to selection principles, to appear in Comment. Math. Univ. Carolin. \\ http://arxiv.org/abs/math.GN/0412498

	$o$-Boundedness of free objects over a Tychonoff space
Author	L.S.Zdomskyy lzdomsky@rambler.ru Ivan Franko Lviv National University,Department of Mechanics and Mathematics
Abstract	In this paper we characterize various sorts of boundedness of the free (abelian) topological group $F(X)$ ($A(X)$) as well as the free locally-convex linear topological space $L(X)$ in terms of properties of a Tychonoff space $X$. These properties appear to be close to so-called selection principles, which permits us to show, that (it is consistent with ZFC that) the property of Hurewicz (Menger) is $l$-invariant. This gives a method of construction of $OF$-undetermined topological groups with strong combinatorial properties.
Keywords	free object, Tychonoff space, topological group
DOI	doi:10.30970/ms.25.1.10-28
Reference	1. Архангельский A.В. Об отображениях, связанных с топологическими группами, ДАН СССР~181 (1968), 1303--1306. 2. Архангельский A.В. О соотношениях между инвариантами топологических групп и их подпространств, Успехи Мат. Наук~35 (1980), 3--22. 3. Архангельский A.В. Топологические пространства функций. -- М.: Изд. Моск. Ун-та, 1989. -- 222~с. 4. Banakh T., On index of total boundedness of (strictly) $o$-bounded groups, Topology Appl. 120 (2002), 427--439. 5. Banakh T., Nikolas P., Sanchis M. Filter games and pathologic subgroups of $\mathbb{R}^\omega$, to appear in J. Aust. Math. Soc.. 6. Banakh T., Zdomsky L. Selection Principles and Games on multicovered spaces and their applications in topological algebra, in preparation. 7. Bartoszynski T., Tsaban B. Hereditary Topological Diagonalizations and the Menger-Hurewicz Conjectures, to appear in Proc. Amer. Math. Soc., \\ http://arxiv.org/abs/math.LO/0208224 8. Bouziad A. Le degr\'e de Lindel\"of est $l$-invariant, Proc. Amer. Math. Soc. 129 (2001), 913--919. 9. Chaber J., Pol R. A remark on Fremlin--Miller theorem concerning the Menger property and Michael concentrated sets, preprint. 10. Энгелькинг Р. Общая топология. -- М.:``Мир'', 1986. -- 751~с. 11. Gerlits J., Nagy Zs. Some properties of $C(X)$, I, Topology Appl. $\mathbf{14}$ (2) (1982), 151--163. 12. Гуран И.И. Топологические группы и свойства их подпространств, Канд. дис., Москва, 1981. -- 69~с. 13. Гуран I.Й., Зарічний M.М. Елементи теорії топологічних груп. -- Київ: УМК ВО, 1991. -- 76~с. 14. Hernandes C. Topological groups close to being $\sigma$-compact, Topology Appl. 102~(2000), 101--111. 15. Hernandes C., Robbie D., Tkachenko M. Some properties of $o$-bounded and strictly $o$-bounded topological groups, Appl. Gen. Topol. 1 (2000), 29--43. 16. Just W., Miller A.~W., Scheepers M., Szeptycki P.~J. The combinatorics of open covers II\/, Topology Appl. 73 (1996), 241--266. 17. Krawczyk A., Michalewski H. An example of a topological group, Topology Appl. 127 (2003), 325--330. 18. Lelek A. Some cover properties of spaces, Fund. Math. LXIV (1969), 209--218. 19. Пестов В.Г. Некоторые свойства свободных топологических групп, Вестн. Москов. Ун-та. 1 (1982), 35--37. 20. Пестов В.Г. Некоторые топологические свойства, сохраняемые отношением $M$-екви\-ва\-лент\-нос\-ти, Успехи Мат. Наук 39 (1984), no.6, 203--304. 21. Pol R. A function space $C(X)$ which is weakly Lindel\"of but not weakly compactly generated, Studia Math. 64 (1979), 279--285. 22. Roelcke W., Dierolf S. Uniform structures on topological groups and their quotients, McGraw-Hill International Book Company, 1981. 23. Scheepers M. A direct proof of a theorem of Telgarsky, Proc. Amer. Math. Soc. 123(11) (1995), 3483--3485. 24. Scheepers M. Combinatorics of open covers I: Ramsey Theory, Topology Appl. 69~(1996), 31--62. 25. Scheepers M., Tsaban B. The combinatorics of Borel covers, Topology Appl. 121~(2002), 357--382. 26. Telgarsky R. Topological games: on the 50th anniversary of the Banach-Mazur game, Rocky Mountain J. Math., 17 (1987), no.3, 227--276. 27. Telgarsky R. On sieve-complete and compact-like spaces, Topology Appl. 16 (1983), 63--68. 28. Telgarsky R. Spaces defined by topological games, Fund. Math. CXVI (1983), 189--207. 29. Telgarsky R. On games of Tops, Math. Scand. 54 (1984), 170--176. 30. Tkachenko M. Introduction to topological groups, Topology Appl. 86 (1998), 179--231. 31. Tkachenko M. Topological groups for topologists: part II, Bol. Soc. Mat. Mexicana 3 (2000), no.3, 1--41. 32. Tkachuk V. Some non-multiplicative properties are $l$-invariant, Comment. Math. Univ. Carolin. 38 (1997), 169--175. 33. Tsaban B. $o$-Bounded groups and other topological groups with strong combinatorial properties, Proc. Amer. Math. Soc., to appear. \\ http://arxiv.org/abs/math.GN/0307225 34. Tsaban B. Selection principles in mathematics: A milestone of open problems, Note Mat. 22 (2003/2004), no. 22, 179--208. \\ http://arxiv.org/abs/math.GN/0312182 35. Tsaban B., Zdomskyy L. Scales, Fields, and a problem of Hurewicz, submitted to J.~Amer. Math. Soc.. http://arxiv.org/abs/math.GN/0507043. 36. Успенский В.В. О топологии свободного локально-выпуклого пространства, ДАН СССР 270 (1983), 1334--1337. 37. Vaughan J. Small uncountable cardinals and topology, in: Open problems in topology (eds. J.~van Mill, G.M.~Reed), Elsevier Sci. Publ., 1990, 197--216. 38. Velichko N.~V. The Lindelof property is $l$-invariant, Topology Appl. 89 (1998), 277--283. 39. Zdomskyy L. A semifilter approach to selection principles, to appear in Comment. Math. Univ. Carolin. \\ http://arxiv.org/abs/math.GN/0412498
Pages	10-28
Volume	25
Issue	1
Year	2006
Journal	Matematychni Studii
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