Can a Borel group be generated by a Hurewicz subspace?

L.S.Zdomskyy

In this paper we formulate three problems concerning topological properties of sets generating Borel non-$\sigma$-compact groups. In the case of a concrete $F_{\sigma\delta}$-subgroup of $\{0,1\}^{\omega\times\omega}$ this gives an equivalent reformulation of the Scheepers diagram problem.

1. Arhangel'skiĭ~A.V. Hurewicz spaces, analytic sets, and fan tightness of function spaces, Soviet Math. Doklady 33 (1986), 396--399.

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

3. Banakh~T., Zdomskyy~L. Separation properties between $\sigma$-compactness and the Hurewicz property, in progress.

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

5. Hurewicz~W. Über Folgen stetiger Functionen, Fund. Math. 9 (1927), 193--204.

6. Just~W., Miller~A., Scheepers~M., Szeptycki~S. The combinatorics of open covers II, Topology Appl. 73 (1996), 241--266.

7. Kechris~A. Classical Descriptive Set Theory. -- Grad. Texts in Math. 156, Springer, 1995.

8. Menger~K. Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte. Abt. 2a, Mathematic, Astronomie, Physic, Meteorologie und Mechanic (Wiener Akademie) 133 (1924), 421--444.

9. Marczewski~E. (Szpilrajn) The characteristic function of a sequence of sets and some of its applications, Fund. Math. 31 (1938), 207--233.

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

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

12. Tsaban~B. Selection principles in mathematics: A milestone of open problems, Note Mat. 22 (2003/2004), no 2, 179--208.

13. Tsaban~B. Some new directions in infinite-combinatorial topology, submitted. http://arxiv.org/abs/math.GN/0409069

14. Tsaban~B. (eds.), SPM Bulletin 2 (2003). http://arxiv.org/abs/math.GN/0302062

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

16. Vaughan~J. Small uncountable cardinals and topology, In: Open Problems in Topology (J. van Mill, G.M. Reed, Eds.), Elsevier Sci. Publ., Amsterdam, 1990, 195--218.

17. Zdomskyy~L. A semifilter approach to selection principles, Comment. Math. Univ. Carolin. 46 (2005), 525--540.

	Can a Borel group be generated by a Hurewicz subspace?
Author	L.S.Zdomskyy lzdomsky@rambler.ru Ivan Franko National University of Lviv
Abstract	In this paper we formulate three problems concerning topological properties of sets generating Borel non-$\sigma$-compact groups. In the case of a concrete $F_{\sigma\delta}$-subgroup of $\{0,1\}^{\omega\times\omega}$ this gives an equivalent reformulation of the Scheepers diagram problem.
Keywords	Borel group, Hurewicz subspace, Scheepers diagram problem
DOI	doi:10.30970/ms.25.2.219-224
Reference	1. Arhangel'skiĭ~A.V. Hurewicz spaces, analytic sets, and fan tightness of function spaces, Soviet Math. Doklady 33 (1986), 396--399. 2. Bartoszyński~T., Tsaban~B. Hereditary Topological Diagonalizations and the Menger-Hurewicz Conjectures, Proc. Amer. Math. Soc., to appear. http://arxiv.org/abs/math.LO/0208224 3. Banakh~T., Zdomskyy~L. Separation properties between $\sigma$-compactness and the Hurewicz property, in progress. 4. Gerlits~J., Nagy~Zs. Some properties of $C(X)$, I, Topology Appl. $\mathbf{14}$ (2) (1982), 151--163. 5. Hurewicz~W. Über Folgen stetiger Functionen, Fund. Math. 9 (1927), 193--204. 6. Just~W., Miller~A., Scheepers~M., Szeptycki~S. The combinatorics of open covers II, Topology Appl. 73 (1996), 241--266. 7. Kechris~A. Classical Descriptive Set Theory. -- Grad. Texts in Math. 156, Springer, 1995. 8. Menger~K. Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte. Abt. 2a, Mathematic, Astronomie, Physic, Meteorologie und Mechanic (Wiener Akademie) 133 (1924), 421--444. 9. Marczewski~E. (Szpilrajn) The characteristic function of a sequence of sets and some of its applications, Fund. Math. 31 (1938), 207--233. 10. Scheepers~M. Combinatorics of open covers I: Ramsey Theory, Topology Appl. 69 (1996), 31--62. 11. Scheepers~M., Tsaban~B. The combinatorics of Borel covers, Topology Appl. 121 (2002), 357--382. 12. Tsaban~B. Selection principles in mathematics: A milestone of open problems, Note Mat. 22 (2003/2004), no 2, 179--208. 13. Tsaban~B. Some new directions in infinite-combinatorial topology, submitted. http://arxiv.org/abs/math.GN/0409069 14. Tsaban~B. (eds.), SPM Bulletin 2 (2003). http://arxiv.org/abs/math.GN/0302062 15. Tsaban~B., Zdomskyy~L. Scales, fields, and a problem of Hurewicz, submitted to J. Amer. Math. Soc. http://arxiv.org/abs/math.GN/0507043. 16. Vaughan~J. Small uncountable cardinals and topology, In: Open Problems in Topology (J. van Mill, G.M. Reed, Eds.), Elsevier Sci. Publ., Amsterdam, 1990, 195--218. 17. Zdomskyy~L. A semifilter approach to selection principles, Comment. Math. Univ. Carolin. 46 (2005), 525--540.
Pages	219-224
Volume	25
Issue	2
Year	2006
Journal	Matematychni Studii
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