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nonseparable metric spaces; Luzin spaces; $\sigma $-discrete network; uniformization; bimeasurable maps
We relate some subsets $G$ of the product $X\times Y$ of nonseparable Luzin (e.g., completely metrizable) spaces to subsets $H$ of $\mathbb{N}^{\mathbb{N}}\times Y$ in a way which allows to deduce descriptive properties of $G$ from corresponding theorems on $H$. As consequences we prove a nonseparable version of Kondô’s uniformization theorem and results on sets of points $y$ in $Y$ with particular properties of fibres $f^{-1}(y)$ of a mapping $f\: X\rightarrow Y$. Using these, we get descriptions of bimeasurable mappings between nonseparable Luzin spaces in terms of fibres.
[1] Z. Frolík: A measurable map with analytic domain and metrizable range is quotient. Bull. Amer. Math. Soc. 76 (1970), 1112–1117. DOI 10.1090/S0002-9904-1970-12584-8 | MR 0265539
[2] Z. Frolík and P. Holický: Applications of Luzinian separation principles (non-separable case). Fund. Math. 117 (1983), 165–185. MR 0719837
[3] R. W. Hansell: On characterizing non-separable analytic and extended Borel sets as types of continuous images. Proc. London Math. Soc. 28 (1974), 683–699. MR 0362269 | Zbl 0313.54044
[4] R. W. Hansell: Descriptive sets and the topology of nonseparable Banach spaces. Serdica Math. J. 27 (2001), 1–66. MR 1828793 | Zbl 0982.46012
[5] R. W. Hansell: Descriptive topology. Recent Progress in General Topology, M. Hušek and J. van Mill (eds.), North-Holland, Amsterdam, London, New York, Tokyo, 1992, pp. 275–315. MR 1229129 | Zbl 0805.54036
[6] P. Holický: Čech analytic and almost $K$-descriptive spaces. Czech. Math. J. 43 (1993), 451–466. MR 1249614
[7] P. Holický: Luzin theorems for scattered-$K$-analytic spaces and Borel measures on them. Atti Sem. Mat. Fis. Univ. Modena XLIV (1996), 395–413. MR 1428772
[8] P. Holický: Generalized analytic spaces, completeness and fragmentability. Czech. Math. J. 51 (2001), 791–818. DOI 10.1023/A:1013769030260 | MR 1864043
[9] P. Holický and V. Komínek: On projections of nonseparable Souslin and Borel sets along separable spaces. Acta Univ. Carolin. Math. Phys. 42 (2001), 33–41. MR 1900390
[10] P. Holický and J. Pelant: Internal descriptions of absolute Borel classes. Topology Appl. 141 (2004), 87–104. MR 2058682
[11] P. Holický and M. Zelený: A converse of the Arsenin-Kunugui theorem on Borel sets with $\sigma $-compact sections. Fund. Math. 165 (2000), 191–202. MR 1805424
[12] J. E. Jayne and C. A. Rogers: Upper semicontinuous set-valued functions. Acta Math. 149 (1982), 87–125. DOI 10.1007/BF02392351 | MR 0674168
[13] J. Kaniewski and R. Pol: Borel-measurable selectors for compact-valued mappings in the non-separable case. Bull. Pol. Acad. Sci. Math. 23 (1975), 1043–1050. MR 0410657
[14] A. S. Kechris: Classical Descriptive Set Theory. Springer, New York etc., 1995. MR 1321597 | Zbl 0819.04002
[15] V. Komínek: A remark on the uniformization in metric spaces. Acta Univ. Carolin. Math. Phys. 40 (1999), 65–74. MR 1751542
[16] R. Purves: Bimeasurable functions. Fund. Math. 58 (1966), 149–157. MR 0199339 | Zbl 0143.07101
[17] C. A. Rogers and R. C. Willmott: On the uniformization of sets in topological spaces. Acta Math. 120 (1968), 1–52. DOI 10.1007/BF02394605 | MR 0237733
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