A game characterization of
limit-detecting sequences in locally compact $G$-spaces

T. O. Banakh; S. I. Pidkuyko

It is proved that under some mild conditions a set $S\subset X$ is strongly limit-detecting if and only if $S$ is $\omega$-controlling in $X$ if and only if $S$ is asymptotically dense in the sense that for any neighborhood $U$ of the unit in $G$ the set $US$ has bounded complement in $X$. On the other hand, $S\subset X$ is limit-detecting if and only if $S$ is 1-controlling and splittable. In its turn, a~set $S\subset X$ is 1-controlling if the product $KS$ is $\omega$-controlling for some compact countable set $K\subset G$. These results are proved with help of some infinite game resembling the Telg\'arski game characterizing $\mathcal K$-scattered properties.

1. Банах Т., Підкуйко С., Здомський Л. Скільки збіжних послідовностей необхідно для виявлення розриву функції в точці, Наукові записки НТШ, 2004.

2. Banakh T., Protasov I. Color-detectors of hypergraphs, in preparation.

3. Blass A. Combinatorial cardinal characteristics of the continuum, in: Handbook of Set Theory (eds.: Foreman M., Kanamori A., Magidor M.), Kluwer, to appear.

4. Дороговцев А. Я. Математический аналиы: Сборник задач, К.: Вища школа, 1987.

5. Энгелькинг Р. Общая топология, М.: Мир, 1986.

6. Голузин М.Г., Лодкин А.А., Макаров Б.М., Подкорытов А.Н. Задачи по математическому анализу, Л.: Изд-во Ленингр. ун-та, 1983.

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

8. Habala P., Hájek P., Zizler V. Introduction to Banach spaces, Praha: Mathfyzpress, 1996.

9. Kechris A. Classical descriptive set theory, Springer-Verlag, 1995.

10. Leif M. Om ligelig kontinuitet i uendelig, Nordisk Math. Tidskr. 24 (1976), Hf. 2, 71–74.

11. Protasov I., Banakh T. Ball structures and colorings of graphs and groups, Mathemetical Studies, Monograph Series, Vol.11. – VNTL: Lviv, 2003.

12. Telgárski R. Spaces defined by topological games, Fund. math. 88 (1975), 193–223.

13. Telgárski R. Topological games: on the 50th anniversary of the Banach-Mazur game, Rocky Mountain J. Math. 17 (1987), no 2, 227–276.

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

15. Ткачук В.В. Топологические приложения теории игр, М.: Изд-во МГУ, 1992.

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

17. Yajima Y. Topological games and applications, in: Topics in General Topology (K.Morita, J.Nagata, Eds.), Elsevier Publ., 1989.

	A game characterization of limit-detecting sequences in locally compact $G$-spaces
Author	T. O. Banakh, S. I. Pidkuyko tbanakh@franko.lviv.ua, pidkyuko@farlep.lviv.ua Lviv Ivan Franko National University
Abstract	It is proved that under some mild conditions a set $S\subset X$ is strongly limit-detecting if and only if $S$ is $\omega$-controlling in $X$ if and only if $S$ is asymptotically dense in the sense that for any neighborhood $U$ of the unit in $G$ the set $US$ has bounded complement in $X$. On the other hand, $S\subset X$ is limit-detecting if and only if $S$ is 1-controlling and splittable. In its turn, a~set $S\subset X$ is 1-controlling if the product $KS$ is $\omega$-controlling for some compact countable set $K\subset G$. These results are proved with help of some infinite game resembling the Telg\'arski game characterizing $\mathcal K$-scattered properties.
Keywords	Telgarski game, asymptotically dense set, strongly limit-detecting set
DOI	doi:10.30970/ms.21.2.115-132
Reference	1. Банах Т., Підкуйко С., Здомський Л. Скільки збіжних послідовностей необхідно для виявлення розриву функції в точці, Наукові записки НТШ, 2004. 2. Banakh T., Protasov I. Color-detectors of hypergraphs, in preparation. 3. Blass A. Combinatorial cardinal characteristics of the continuum, in: Handbook of Set Theory (eds.: Foreman M., Kanamori A., Magidor M.), Kluwer, to appear. 4. Дороговцев А. Я. Математический аналиы: Сборник задач, К.: Вища школа, 1987. 5. Энгелькинг Р. Общая топология, М.: Мир, 1986. 6. Голузин М.Г., Лодкин А.А., Макаров Б.М., Подкорытов А.Н. Задачи по математическому анализу, Л.: Изд-во Ленингр. ун-та, 1983. 7. Guran I. On topological groups close to being Lindelöf, Soviet Math. Dokl. 23 (1981), 173–175. 8. Habala P., Hájek P., Zizler V. Introduction to Banach spaces, Praha: Mathfyzpress, 1996. 9. Kechris A. Classical descriptive set theory, Springer-Verlag, 1995. 10. Leif M. Om ligelig kontinuitet i uendelig, Nordisk Math. Tidskr. 24 (1976), Hf. 2, 71–74. 11. Protasov I., Banakh T. Ball structures and colorings of graphs and groups, Mathemetical Studies, Monograph Series, Vol.11. – VNTL: Lviv, 2003. 12. Telgárski R. Spaces defined by topological games, Fund. math. 88 (1975), 193–223. 13. Telgárski R. Topological games: on the 50th anniversary of the Banach-Mazur game, Rocky Mountain J. Math. 17 (1987), no 2, 227–276. 14. Tkachenko M. Introduction to topological groups, Topology Appl. 86 (1998), 179–231. 15. Ткачук В.В. Топологические приложения теории игр, М.: Изд-во МГУ, 1992. 16. Vaughan J.E. Small uncountable cardinals and topology, in: J. van Mill, G.M. Reed (eds). Open problems in topology, (Elsevier Sci. Publ., 1990) 197–216. 17. Yajima Y. Topological games and applications, in: Topics in General Topology (K.Morita, J.Nagata, Eds.), Elsevier Publ., 1989.
Pages	115-132
Volume	21
Issue	2
Year	2004
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

A game characterization of limit-detecting sequences in locally compact $G$-spaces