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Absolute Borel sets and function spaces. (English) Zbl 0880.54023
Summary: An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolík for characterizing Čech-complete spaces. We also show that the absolute Borel class of $$X$$ is determined by the uniform structure of the space of continuous functions $$C_{p}(X)$$; however the case of absolute $$G_{\delta}$$ metric spaces is still open. More precisely, we prove that, for metrizable spaces $$X$$ and $$Y$$, if $$\Phi: C_{p}(X) \rightarrow C_{p}(Y)$$ is a uniformly continuous surjection and $$X$$ is an absolute Borel set of multiplicative (resp., additive) class $$\alpha$$, $$\alpha>1$$, then $$Y$$ is also an absolute Borel set of the same class. This result is new even if $$\Phi$$ is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the Čech-completeness of a metric space $$X$$ is determined by the linear structure of $$C_{p}(X)$$.

##### MSC:
 54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) 54C35 Function spaces in general topology 03E15 Descriptive set theory 54E35 Metric spaces, metrizability
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##### References:
 [1] A. V. Arkhangel$$^{\prime}$$skiĭ, Topological function spaces, Mathematics and its Applications (Soviet Series), vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the Russian by R. A. M. Hoksbergen. [2] A. V. Arhangel$$^{\prime}$$skiĭ, On topological spaces which are complete in the sense of Čech, Vestnik Moskov. Univ. Ser. I Mat. Meh. 1961 (1961), no. 2, 37 – 40 (Russian, with English summary). [3] Jan Baars, Joost de Groot, and Jan Pelant, Function spaces of completely metrizable spaces, Trans. Amer. Math. Soc. 340 (1993), no. 2, 871 – 883. · Zbl 0841.54012 [4] Alberto Barbati, The hyperspace of an analytic metrizable space is analytic, Proceedings of the Eleventh International Conference of Topology (Trieste, 1993), 1993, pp. 15 – 21 (1994) (English, with English and Italian summaries). · Zbl 0849.54030 [5] Gerald Beer, Topologies on closed and closed convex sets, Mathematics and its Applications, vol. 268, Kluwer Academic Publishers Group, Dordrecht, 1993. · Zbl 0792.54008 [6] Czesław Bessaga and Aleksander Pełczyński, Selected topics in infinite-dimensional topology, PWN — Polish Scientific Publishers, Warsaw, 1975. Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58]. · Zbl 0304.57001 [7] J. P. R. Christensen, Topology and Borel structure, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. Descriptive topology and set theory with applications to functional analysis and measure theory; North-Holland Mathematics Studies, Vol. 10. (Notas de Matemática, No. 51). · Zbl 0273.28001 [8] C. Costantini, Every Wijsman topology relative to a Polish space is Polish, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2569 – 2574. · Zbl 0831.54014 [9] C. Costantini, S. Levi, J. Pelant, On compactness in hyperspaces, in preparation. · Zbl 1024.54005 [10] T. Dobrowolski, S. P. Gul$$^{\prime}$$ko, and J. Mogilski, Function spaces homeomorphic to the countable product of \?^\?$$_{2}$$, Topology Appl. 34 (1990), no. 2, 153 – 160. · Zbl 0691.57009 · doi:10.1016/0166-8641(90)90077-F · doi.org [11] Tadeusz Dobrowolski and Witold Marciszewski, Classification of function spaces with the pointwise topology determined by a countable dense set, Fund. Math. 148 (1995), no. 1, 35 – 62. · Zbl 0834.46016 [12] Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna. Tom 47. [Mathematics Library. Vol. 47]. Ryszard Engelking, General topology, PWN — Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. [13] D. H. Fremlin, Families of compact sets and Tukey’s ordering, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), no. 1, 29 – 50. · Zbl 0772.54030 [14] Zdeněk Frolík, Generalizations of the \?_\?-property of complete metric spaces, Czechoslovak Math. J 10 (85) (1960), 359 – 379 (English, with Russian summary). · Zbl 0100.18701 [15] -, Topologically complete spaces, Comment. Math. Univ. Carol. 1 (1960), 1-3. · Zbl 0100.18702 [16] Z. Frolík, A contribution to the descriptive theory of sets and spaces, General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961) Academic Press, New York; Publ. House Czech. Acad. Sci., Prague, 1962, pp. 157 – 173. [17] Zdeněk Frolík, A survey of separable descriptive theory of sets and spaces, Czechoslovak Math. J. 20 (95) (1970), 406 – 467. · Zbl 0223.54028 [18] Sergei Gul’ko, The space \?_\?(\?) for countable infinite compact \? is uniformly homeomorphic to \?$$_{0}$$, Bull. Polish Acad. Sci. Math. 36 (1988), no. 5-6, 391 – 396 (1989) (English, with Russian summary). · Zbl 0754.54010 [19] S. P. Gul$$^{\prime}$$ko, On uniform homeomorphisms of spaces of continuous functions, Trudy Mat. Inst. Steklov. 193 (1992), 82 – 88 (Russian); English transl., Proc. Steklov Inst. Math. 3(193) (1993), 87 – 93. [20] R. W. Hansell, Descriptive topology, Recent progress in general topology, M. Husek and J. van Mill, editors, North-Holland, Amsterdam, 1992, pp. 275-315. CMP 93:15 [21] J. R. Isbell, Uniform spaces, Mathematical Surveys, No. 12, American Mathematical Society, Providence, R.I., 1964. · Zbl 0124.15601 [22] H. J. K. Junnila and H. P. A. Künzi, Characterizations of absolute $$F_{\sigma \delta }$$-sets, preprint. · Zbl 0926.54018 [23] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. · Zbl 0819.04002 [24] Victor Klee, On the Borelian and projective types of linear subspaces, Math. Scand. 6 (1958), 189 – 199. · Zbl 0088.08502 · doi:10.7146/math.scand.a-10543 · doi.org [25] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. · Zbl 0158.40901 [26] O. G. Okunev, Weak topology of a dual space and a \?-equivalence relation, Mat. Zametki 46 (1989), no. 1, 53 – 59, 123 (Russian); English transl., Math. Notes 46 (1989), no. 1-2, 534 – 538 (1990). · Zbl 0774.46020 · doi:10.1007/BF01159103 · doi.org [27] Jean Saint-Raymond, La structure borélienne d’Effros est-elle standard?, Fund. Math. 100 (1978), no. 3, 201 – 210 (French). · Zbl 0434.54028 [28] W. Sierpinski, Sur une définition topologique des ensembles $$F_{\sigma \delta }$$, Fund. Math. 6 (1924), 24-29. [29] V. V. Uspenskiĭ, A characterization of compactness in terms of the uniform structure in a space of functions, Uspekhi Mat. Nauk 37 (1982), no. 4(226), 183 – 184 (Russian). [30] V. Valov, Linear mappings between function spaces, preprint. · Zbl 0885.54012
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