𝐹_{𝜎}–additive families and the invariance of Borel classes

Spurný, Jiří

doi:10.1090/S0002-9939-04-07587-2

$F_\sigma$–additive families and the invariance of Borel classes
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Proc. Amer. Math. Soc. 133 (2005), 905-915 Request permission

Abstract:

We prove that any $F_\sigma$–additive family $\mathcal {A}$ of sets in an absolutely Souslin metric space has a $\sigma$–discrete refinement provided every partial selector set for $\mathcal {A}$ is $\sigma$–discrete. As a corollary we obtain that every mapping of a metric space onto an absolutely Souslin metric space, which maps $F_\sigma$–sets to $F_\sigma$–sets and has complete fibers, admits a section of the first class. The invariance of Borel and Souslin sets under mappings with complete fibers, which preserves $F_\sigma$-sets, is shown as an application of the previous result.

References

A. G. El′kin, $A$-sets in complete metric spaces, Dokl. Akad. Nauk SSSR 175 (1967), 517–520 (Russian). MR 0214029
Siegfried Graf, Selected results on measurable selections, Rend. Circ. Mat. Palermo (2) Suppl. 2 (1982), 87–122. MR 683772
R. W. Hansell, Borel measurable mappings for nonseparable metric spaces, Trans. Amer. Math. Soc. 161 (1971), 145–169. MR 288228, DOI 10.1090/S0002-9947-1971-0288228-1
R. W. Hansell, $F_\sigma$-set covers of analytic spaces and first class selectors, Proc. Amer. Math. Soc. 96 (1986), no. 2, 365–371. MR 818473, DOI 10.1090/S0002-9939-1986-0818473-1
R. W. Hansell, Nonseparable analytic metric spaces and quotient maps, Topology Appl. 85 (1998), no. 1-3, 143–152. 8th Prague Topological Symposium on General Topology and Its Relations to Modern Analysis and Algebra (1996). MR 1617459, DOI 10.1016/S0166-8641(97)00146-6
R. W. Hansell, On characterizing non-separable analytic and extended Borel sets as types of continuous images, Proc. London Math. Soc. (3) 28 (1974), 683–699. MR 362269, DOI 10.1112/plms/s3-28.4.683
Petr Holický and Jiří Spurný, Perfect images of absolute Souslin and absolute Borel Tychonoff spaces, Topology Appl. 131 (2003), no. 3, 281–294. MR 1983084, DOI 10.1016/S0166-8641(02)00356-5
J. E. Jayne and C. A. Rogers, Invariance of Borel classes in metric spaces, Math. Ann. 263 (1983), no. 3, 323–341. MR 704298, DOI 10.1007/BF01457135
J. Kaniewski and R. Pol, Borel-measurable selectors for compact-valued mappings in the non-separable case, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), no. 10, 1043–1050 (English, with Russian summary). MR 410657
George Koumoullis, Cantor sets in Prohorov spaces, Fund. Math. 124 (1984), no. 2, 155–161. MR 774507, DOI 10.4064/fm-124-2-155-161
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751