𝐹_{𝜎}-set covers of analytic spaces and first class selectors

Hansell, R. W.

doi:10.1090/S0002-9939-1986-0818473-1

$F_ \sigma$-set covers of analytic spaces and first class selectors
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by R. W. Hansell

Proc. Amer. Math. Soc. 96 (1986), 365-371

DOI: https://doi.org/10.1090/S0002-9939-1986-0818473-1

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Abstract:

Let $X$ be an analytic space (e.g., a complete metric space). We prove that any point-countable ${F_\sigma }$-set cover of $X$ either has $\sigma$-discrete refinement, or else there is a compact subset of $X$ not covered by any countable subfamily of the cover. It follows that any point-countable ${F_\sigma }$-additive family in $X$ has a $\sigma$-discrete refinement. This is used to show that any weakly ${F_\sigma }$-measurable multimap, defined on $X$ and taking nonempty, closed and separable values in a complete metric space, has a selector of the first Baire class.

References

A. G. El′kin, $A$-sets in complete metric spaces, Dokl. Akad. Nauk SSSR 175 (1967), 517–520 (Russian). MR 0214029
William G. Fleissner, An axiom for nonseparable Borel theory, Trans. Amer. Math. Soc. 251 (1979), 309–328. MR 531982, DOI 10.1090/S0002-9947-1979-0531982-9
William G. Fleissner, Roger W. Hansell, and Heikki J. K. Junnila, PMEA implies proposition $\textrm {P}$, Topology Appl. 13 (1982), no. 3, 255–262. MR 651508, DOI 10.1016/0166-8641(82)90034-7
D. H. Fremlin, Measure-additive coverings and measurable selectors, Dissertationes Math. (Rozprawy Mat.) 260 (1987), 116. MR 928693
Zdeněk Frolík and Petr Holický, Decomposability of completely Suslin-additive families, Proc. Amer. Math. Soc. 82 (1981), no. 3, 359–365. MR 612719, DOI 10.1090/S0002-9939-1981-0612719-6
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, Generalized quotient maps that are inductively index-$\sigma$-discrete, Pacific J. Math. 117 (1985), no. 1, 99–119. MR 777439, DOI 10.2140/pjm.1985.117.99

A measurable selection and representation theorem in non-separable spaces

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
R. W. Hansell, J. E. Jayne, and C. A. Rogers, $K$-analytic sets, Mathematika 30 (1983), no. 2, 189–221 (1984). MR 737176, DOI 10.1112/S0025579300010524

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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
E. Michael, On maps related to $\sigma$-locally finite and $\sigma$-discrete collections of sets, Pacific J. Math. 98 (1982), no. 1, 139–152. MR 644945, DOI 10.2140/pjm.1982.98.139
Roman Pol, Note on decompositions of metrizable spaces. II, Fund. Math. 100 (1978), no. 2, 129–143. MR 494011, DOI 10.4064/fm-100-2-129-143
A. H. Stone, On $\sigma$-discreteness and Borel isomorphism, Amer. J. Math. 85 (1963), 655–666. MR 156789, DOI 10.2307/2373113
A. H. Stone, Kernel constructions and Borel sets, Trans. Amer. Math. Soc. 107 (1963), 58–70; errata: 107 (1963), 558. MR 0151935, DOI 10.1090/S0002-9947-1963-0151935-0
A. H. Stone, Non-separable Borel sets. II, General Topology and Appl. 2 (1972), 249–270. MR 312481, DOI 10.1016/0016-660X(72)90010-4