A characterization of the Menger and Hurewicz properties of subspaces of the real line

L.S.Zdomskyy

We characterize the covering properties of Menger, Hurewicz, and two other selection principles of subsets of the real line in terms of their continuous images in the Baire space $\mathbb N^\omega$, and thus answer the corresponding question of B. Tsaban in positive.

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

2. Hurewicz W. Über Folgen stetiger Functionen, Fundamenta Mathematicae 9 (1927), 193--204;

3. Келли Дж. Л. Общая топология. -- М.: Наука, 1968.

4. Куратовский К. Топология. -- М.: Мир, 1969.

5. Gerlits J., Nagy Zs. Some properties of $C(X)$, I, Topol. Appl. $\mathbf{14}$ (1982), no.2, 151--163.

6. Архангельський A. В. Пространства Гуревича, аналитические множества и веерная теснота пространств функций, ДАН СССР, 287 (1986), № 3, 525--528.

7. Reclaw I. Every Luzin set is undetermined in the point-open game, Fundamenta Mathematicae 144 (1994), 43--54.

8. Kechris A. Classical Descriptive Set Theory, Springer, 1995.

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

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

11. Tsaban B. A diagonalization property between Hurewicz and Menger, Real Analysis Exchange, 27 (2001/2002), no.2, 1--7.

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

13. Tsaban B. The Hurewicz covering property and slaloms in the Baire space, Fundamenta Mathematicae 181 (2004), 273--280.

14. Tsaban B. Selection principles and the minimal tower problem, to appear in Note di Matematica, http://arxiv.org/abs/math.LO/0105045.

	A characterization of the Menger and Hurewicz properties of subspaces of the real line
Author	L.S.Zdomskyy lzdomsky@rambler.ru Ivan Franko National University of Lviv
Abstract	We characterize the covering properties of Menger, Hurewicz, and two other selection principles of subsets of the real line in terms of their continuous images in the Baire space $\mathbb N^\omega$, and thus answer the corresponding question of B. Tsaban in positive.
Keywords	Menger property, Hurewicz propertty, Baire space
DOI	doi:10.30970/ms.24.2.115-119
Reference	1. Menger K. Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte. Abt. 2a, Mathematic, Astronomie, Physic, Meteorologie und Mechanic (Wiener Akademie) 133(1924), 421--444. 2. Hurewicz W. Über Folgen stetiger Functionen, Fundamenta Mathematicae 9 (1927), 193--204; 3. Келли Дж. Л. Общая топология. -- М.: Наука, 1968. 4. Куратовский К. Топология. -- М.: Мир, 1969. 5. Gerlits J., Nagy Zs. Some properties of $C(X)$, I, Topol. Appl. $\mathbf{14}$ (1982), no.2, 151--163. 6. Архангельський A. В. Пространства Гуревича, аналитические множества и веерная теснота пространств функций, ДАН СССР, 287 (1986), № 3, 525--528. 7. Reclaw I. Every Luzin set is undetermined in the point-open game, Fundamenta Mathematicae 144 (1994), 43--54. 8. Kechris A. Classical Descriptive Set Theory, Springer, 1995. 9. Scheepers M. Combinatorics of open covers I: Ramsey Theory, Topology and Appl. 69 (1996), 31--62. 10. Just W., Miller A.W., Scheepers M., Szeptycki P.J. The combinatorics of open covers II\/, Topology and Appl. 73 (1996), 241--266. 11. Tsaban B. A diagonalization property between Hurewicz and Menger, Real Analysis Exchange, 27 (2001/2002), no.2, 1--7. 12. Scheepers M., Tsaban B. The combinatorics of Borel covers, Topology and Appl. 121 (2002), 357--382. 13. Tsaban B. The Hurewicz covering property and slaloms in the Baire space, Fundamenta Mathematicae 181 (2004), 273--280. 14. Tsaban B. Selection principles and the minimal tower problem, to appear in Note di Matematica, http://arxiv.org/abs/math.LO/0105045.
Pages	115-119
Volume	24
Issue	2
Year	2005
Journal	Matematychni Studii
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