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diagonal of a mapping; separately continuous mapping; Baire-one mapping; equiconnected space; strongly $\sigma$-metrizable space
We prove the result on Baire classification of mappings $f:X\times Y\to Z$ which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where $X$ is a $PP$-space, $Y$ is a topological space and $Z$ is a strongly $\sigma$-metrizable space with additional properties. We show that for any topological space $X$, special equiconnected space $Z$ and a mapping $g:X\to Z$ of the $(n-1)$-th Baire class there exists a strongly separately continuous mapping $f:X^n\to Z$ with the diagonal $g$. For wide classes of spaces $X$ and $Z$ we prove that diagonals of separately continuous mappings $f:X^n\to Z$ are exactly the functions of the $(n-1)$-th Baire class. An example of equiconnected space $Z$ and a Baire-one mapping $g:[0,1]\to Z$, which is not a diagonal of any separately continuous mapping $f:[0,1]^2\to Z$, is constructed.
[1] Baire R.: Sur les fonctions de variables reélles. Ann. Mat. Pura Appl., ser. 3 (1899), no. 3, 1–123.
[2] Banakh T.: (Metrically) Quater-stratifable spaces and their applications in the theory of separately continuous functions. Topology Appl. 157 (2010), no. 1, 10–28. MR 1968755
[3] Burke M.: Borel measurability of separately continuous functions. Topology Appl. 129 (2003), no. 1, 29–65. DOI 10.1016/S0166-8641(02)00136-0 | MR 1955664
[4] Engelking R.: General Topology. revised and completed edition, Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[5] Fosgerau M.: When are Borel functions Baire functions?. Fund. Math. 143 (1993), 137–152. MR 1240630
[6] Hahn H.: Theorie der reellen Funktionen. 1. Band. Springer, Berlin, 1921.
[7] Hansell R.W.: Borel measurable mappings for nonseparable metric spaces. Trans. Amer. Math. Soc. 161 (1971), 145–169. DOI 10.1090/S0002-9947-1971-0288228-1 | MR 0288228
[8] Hansell R.W.: On Borel mappings and Baire functions. Trans. Amer. Math. Soc. 194 (1974), 145–169. DOI 10.1090/S0002-9947-1974-0362270-7 | MR 0362270
[9] Karlova O., Maslyuchenko V., Mykhaylyuk V.: Equiconnected spaces and Baire classification of separately continuous functions and their analogs. Cent. Eur. J. Math., 10 (2012), no. 3, 1042–1053. DOI 10.2478/s11533-012-0016-8 | MR 2902232
[10] Karlova O., Mykhaylyuk V., Sobchuk O.: Diagonals of separately continuous functions and their analogs. Topology Appl. 160 (2013), 1–8. DOI 10.1016/j.topol.2012.09.003 | MR 2995069
[11] Karlova O.: Classification of separately continuous functions with values in sigma-metrizable spaces. Applied Gen. Top. 13 (2012), no. 2, 167–178. MR 2998364
[12] Karlova O.: Functionally $\sigma$-discrete mappings and a generalization of Banach's theorem. Topology Appl. 189 (2015), 92–106. DOI 10.1016/j.topol.2015.04.014 | MR 3342574
[13] Karlova O.: On Baire classification of mappings with values in connected spaces. Eur. J. Math., DOI: 10.1007/s40879-015-0076-y. DOI 10.1007/s40879-015-0076-y
[14] Lebesgue H.: Sur l'approximation des fonctions. Bull. Sci. Math. 22 (1898), 278–287.
[15] Lebesgue H.: Sur les fonctions respresentables analytiquement. Journ. de Math. 2 (1905), no. 1, 139–216.
[16] Maslyuchenko O., Maslyuchenko V., Mykhaylyuk V., Sobchuk O.: Paracompactness and separately continuous mappings. General Topology in Banach Spaces, Nova Sci. Publ., Huntington, New York, 2001, pp. 147–169. MR 1901542
[17] Moran W.: Separate continuity and support of measures. J. London. Math. Soc. 44 (1969), 320–324. DOI 10.1112/jlms/s1-44.1.320 | MR 0236346
[18] Mykhaylyuk V.: Construction of separately continuous functions of $n$ variables with the given restriction. Ukr. Math. Bull. 3 (2006), no. 3, 374–381 (in Ukrainian). MR 2330679
[19] Mykhaylyuk V.: Baire classification of separately continuous functions and Namioka property. Ukr. Math. Bull. 5 (2008), no. 2, 203–218 (in Ukrainian). MR 2559835
[20] Mykhaylyuk V., Sobchuk O., Fotij O.: Diagonals of separately continuous multivalued mappings. Mat. Stud. 39 (2013), no. 1, 93–98 (in Ukrainian). MR 3099603
[21] Rudin W.: Lebesgue's first theorem. Math. Analysis and Applications, Part B. Adv. in Math. Supplem. Studies, 7B (1981), 741–747. MR 0634266
[22] Sobchuk O.: Baire classification and Lebesgue spaces. Sci. Bull. Chernivtsi Univ. 111 (2001), 110–112 (in Ukrainian).
[23] Sobchuk O.: $PP$-spaces and Baire classification. Int. Conf. on Funct. Analysis and its Appl. Dedic. to the 110-th ann. of Stefan Banach (May 28-31, Lviv) (2002), p. 189.
[24] Schaefer H.: Topological Vector Spaces. Macmillan, 1966. MR 0193469 | Zbl 0983.46002
[25] Vera G.: Baire measurability of separately continuous functions. Quart. J. Math. Oxford. 39 (1988), no. 153, 109–116. DOI 10.1093/qmath/39.1.109 | MR 0929799
[26] Veselý L.: Characterization of Baire-one functions between topological spaces. Acta Univ. Carol., Math. Phys. 33 (1992), no. 2, 143–156. MR 1287236
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