On questions connected with Talagrand's problem (in Ukrainian)

V.Mykhaylyuk

We prove the following results.
1. If $X$ is a $\alpha$-favourable space, $Y$ is a regular space, in which every separable closed set is compact, and $f\colon X\times Y\to\mathbb R$ is a separately continuous everywhere jointly discontinuous function, then there exists a subspace $Y_0\subseteq Y$ which is homeomorphic to $\beta\mathbb N$.
2. There exist a $\alpha$-favourable space $X$, a dense in $\beta\mathbb N\setminus\mathbb N$ countably compact space $Y$ and a separately continuous everywhere jointly discontinuous function $f\colon X\times Y\to\mathbb R$. In addition, it was obtained some conditions equivalent to the fact that the space $C_p(\beta\mathbb N\setminus\mathbb N,\{0,1\})$ of all continuous functions $x\colon \beta\mathbb N\setminus\mathbb N\to\{0,1\}$ with the topology of point wise convergence is a Baire space.

1. Baire R. Sur les fonctions de variables réelles// Annali di mat. pura ed appl. ser.3. -- 1899. -- V.3. -- P.1-123.

2. Maslyuchenko O.V., Maslyuchenko V.K., Mykhaylyuk V.V., Sobchuk O.V. Paracompactness and separately continuous mappings// General Topology in Banach Spaces. -- Nova Sci. Publ. -- Nantintong-New-York. -- 2001. -- P.147-169.

3. Namioka I. Separate continuity and joint continuity// Pacif. J. Math. -- 1974. -- V. 51, №2. -- P.515-531.

4. Talagrand M. Espaces de Baire et espaces de Namioka// Ann. of. Math. -- 1985. -- V. 270, №2. -- P.159-164.

5. Маслюченко О.В., Михайлюк В.В. До проблеми Талаґрана// Наук. вiсн. Чернiв. ун-ту. Вип. 46. Математика. - Чернiвцi: ЧДУ, 1999. - С.95-99.

6. Saint-Raymond J. Jeux topologiques et espaces de Namioka// Proc. Amer. Math. Soc. -- 1983. -- V. 87, №3. -- P.489-504.

7. Энгелькинг Р. Общая топология. - М.: Мир, 1986. - 752с.

	On questions connected with Talagrand's problem (in Ukrainian)
Author	V.Mykhaylyuk mathan@ukr.net Yuriy Fedkovych Chernivtsi National University
Abstract	We prove the following results. 1. If $X$ is a $\alpha$-favourable space, $Y$ is a regular space, in which every separable closed set is compact, and $f\colon X\times Y\to\mathbb R$ is a separately continuous everywhere jointly discontinuous function, then there exists a subspace $Y_0\subseteq Y$ which is homeomorphic to $\beta\mathbb N$. 2. There exist a $\alpha$-favourable space $X$, a dense in $\beta\mathbb N\setminus\mathbb N$ countably compact space $Y$ and a separately continuous everywhere jointly discontinuous function $f\colon X\times Y\to\mathbb R$. In addition, it was obtained some conditions equivalent to the fact that the space $C_p(\beta\mathbb N\setminus\mathbb N,\{0,1\})$ of all continuous functions $x\colon \beta\mathbb N\setminus\mathbb N\to\{0,1\}$ with the topology of point wise convergence is a Baire space.
Keywords	Talagrand problem, separately continuous everywhere jointly discontinuous function, Baire space, regular space
DOI	doi:10.30970/ms.29.1.81-88
Reference	1. Baire R. Sur les fonctions de variables réelles// Annali di mat. pura ed appl. ser.3. -- 1899. -- V.3. -- P.1-123. 2. Maslyuchenko O.V., Maslyuchenko V.K., Mykhaylyuk V.V., Sobchuk O.V. Paracompactness and separately continuous mappings// General Topology in Banach Spaces. -- Nova Sci. Publ. -- Nantintong-New-York. -- 2001. -- P.147-169. 3. Namioka I. Separate continuity and joint continuity// Pacif. J. Math. -- 1974. -- V. 51, №2. -- P.515-531. 4. Talagrand M. Espaces de Baire et espaces de Namioka// Ann. of. Math. -- 1985. -- V. 270, №2. -- P.159-164. 5. Маслюченко О.В., Михайлюк В.В. До проблеми Талаґрана// Наук. вiсн. Чернiв. ун-ту. Вип. 46. Математика. - Чернiвцi: ЧДУ, 1999. - С.95-99. 6. Saint-Raymond J. Jeux topologiques et espaces de Namioka// Proc. Amer. Math. Soc. -- 1983. -- V. 87, №3. -- P.489-504. 7. Энгелькинг Р. Общая топология. - М.: Мир, 1986. - 752с.
Pages	81-88
Volume	29
Issue	1
Year	2008
Journal	Matematychni Studii
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