Baire classification and $\sigma$-metrizable spaces (in Ukrainian)

V.K.Maslyuchenko; O.V.Sobchuk

It is considered the ascending to Lebesque problem: for what topological spaces $X$ and $Y$ every separately continuous function $f:X\t Y\to\IR$ belongs to the first Baire class. The main theorem develops one of Rudin's result: if $X$ is $\sigma$-metrizable, $Y$ is perfectly normal spaces such that $X\t Y$ is perfectly normal, and $f:X\t Y\to\IR$ is continuous on the first and class $\a$ on the second variable function, then $f$ is of the class $\a+1$ on joint variables. This result is applied to the investigation of separately continuous functions on the products of some strict inductive limits and locally convex spaces in the weak topology.

1. Энгелькинг Р. Общая топология.-- М.: Мир, 1986.-- 751с.

2. Бурбаки Н. Топологические векторные постранства.-- М.: Изд-во иностр. лит., 1951. -- 410 с.

3. Rudin W. {\it Lebesque first theorem} // Math. Analysis and applications, Part B. Edited by L.~Nachbin. Adv. in Math. supplem. studies 7B. Academic Press. -- 1981. -- P.741--747.

4. Moran W. {\it Separate continuity and support of measures}// J. London. Math. Soc. -- 1969. -- 44. -- P.320--324.

5. Vera G. {\it Baire measurability of separately continuous functions}// Quart. J. Math. Oxford. (2). -- 1988. -- 39. -- P.109--116.

6. Собчук О.В. {\it Нарiзно неперервнi функцiї на просторi фiнiтних послiдовностей}. -- Чернiвцi, 1993. -- 5 с. -- Деп. в ДНТБ України, N1701--Ук93.

7. Маслюченко В.К., Собчук О.В. {\it Досконала нормальнiсть простору фiнiтних послiдовностей.} -- Чернiвцi, 1991. -- 6 с. -- Деп. в УкрНДIНТI, N1610--Ук91.

8. Talagrand M. {\it Sur une conjecture de H.H. Corson}// Bull. Soc. Math. -- 1975. -- 99, N4 -- P.211-212.

9. Morita K. {\it On the product of a normal space with a metric space}// Proc. Japan Acad. -- 1963. -- 39. -- P.148--150.

10. Banakh T. {\it Topology of inductive limits of locally convex spaces,} preprint, 1993.

11. Данфорд Н., Шварц Дж. Линейные операторы. Общая теория. -- М.: Изд-во иностр. л-ры, 1962. -- 895 с.

12. Mankiewicz P. {\it On topological Lipschitz and uniform classification of LF-spaces}// Studia Math. -- 1974. -- 52. -- P.109--142.

	Baire classification and $\sigma$-metrizable spaces (in Ukrainian)
Author	V.K.Maslyuchenko, O.V.Sobchuk Department of Mathematics, Chernivtsi University, Kotsyubynska 2, Chernivtsi, 274000, Ukraine
Abstract	It is considered the ascending to Lebesque problem: for what topological spaces $X$ and $Y$ every separately continuous function $f:X\t Y\to\IR$ belongs to the first Baire class. The main theorem develops one of Rudin's result: if $X$ is $\sigma$-metrizable, $Y$ is perfectly normal spaces such that $X\t Y$ is perfectly normal, and $f:X\t Y\to\IR$ is continuous on the first and class $\a$ on the second variable function, then $f$ is of the class $\a+1$ on joint variables. This result is applied to the investigation of separately continuous functions on the products of some strict inductive limits and locally convex spaces in the weak topology.
Keywords	Baire classification, metrizable space, perfectly normal space
DOI	doi:10.30970/ms.3.1.95-102
Reference	1. Энгелькинг Р. Общая топология.-- М.: Мир, 1986.-- 751с. 2. Бурбаки Н. Топологические векторные постранства.-- М.: Изд-во иностр. лит., 1951. -- 410 с. 3. Rudin W. {\it Lebesque first theorem} // Math. Analysis and applications, Part B. Edited by L.~Nachbin. Adv. in Math. supplem. studies 7B. Academic Press. -- 1981. -- P.741--747. 4. Moran W. {\it Separate continuity and support of measures}// J. London. Math. Soc. -- 1969. -- 44. -- P.320--324. 5. Vera G. {\it Baire measurability of separately continuous functions}// Quart. J. Math. Oxford. (2). -- 1988. -- 39. -- P.109--116. 6. Собчук О.В. {\it Нарiзно неперервнi функцiї на просторi фiнiтних послiдовностей}. -- Чернiвцi, 1993. -- 5 с. -- Деп. в ДНТБ України, N1701--Ук93. 7. Маслюченко В.К., Собчук О.В. {\it Досконала нормальнiсть простору фiнiтних послiдовностей.} -- Чернiвцi, 1991. -- 6 с. -- Деп. в УкрНДIНТI, N1610--Ук91. 8. Talagrand M. {\it Sur une conjecture de H.H. Corson}// Bull. Soc. Math. -- 1975. -- 99, N4 -- P.211-212. 9. Morita K. {\it On the product of a normal space with a metric space}// Proc. Japan Acad. -- 1963. -- 39. -- P.148--150. 10. Banakh T. {\it Topology of inductive limits of locally convex spaces,} preprint, 1993. 11. Данфорд Н., Шварц Дж. Линейные операторы. Общая теория. -- М.: Изд-во иностр. л-ры, 1962. -- 895 с. 12. Mankiewicz P. {\it On topological Lipschitz and uniform classification of LF-spaces}// Studia Math. -- 1974. -- 52. -- P.109--142.
Pages	95-102
Volume	3
Issue	1
Year	1994
Journal	Matematychni Studii
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