## Absolute Borel sets and function spaces

HTML articles powered by AMS MathViewer

- by Witold Marciszewski and Jan Pelant PDF
- Trans. Amer. Math. Soc.
**349**(1997), 3585-3596 Request permission

## Abstract:

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 $C_{p}(X)$.## References

- A. V. Arkhangel′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. MR**1144519**, DOI 10.1007/978-94-011-2598-7 - A. V. Arhangel′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). MR**0131258** - Jan Baars, Joost de Groot, and Jan Pelant,
*Function spaces of completely metrizable spaces*, Trans. Amer. Math. Soc.**340**(1993), no. 2, 871–883. MR**1160154**, DOI 10.1090/S0002-9947-1993-1160154-X - 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). MR**1346314** - Gerald Beer,
*Topologies on closed and closed convex sets*, Mathematics and its Applications, vol. 268, Kluwer Academic Publishers Group, Dordrecht, 1993. MR**1269778**, DOI 10.1007/978-94-015-8149-3 - Czesław Bessaga and Aleksander Pełczyński,
*Selected topics in infinite-dimensional topology*, Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58], PWN—Polish Scientific Publishers, Warsaw, 1975. MR**0478168** - J. P. R. Christensen,
*Topology and Borel structure*, North-Holland Mathematics Studies, Vol. 10, 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. MR**0348724** - C. Costantini,
*Every Wijsman topology relative to a Polish space is Polish*, Proc. Amer. Math. Soc.**123**(1995), no. 8, 2569–2574. MR**1273484**, DOI 10.1090/S0002-9939-1995-1273484-5 - C. Costantini, S. Levi, J. Pelant,
*On compactness in hyperspaces*, in preparation. - T. Dobrowolski, S. P. Gul′ko, and J. Mogilski,
*Function spaces homeomorphic to the countable product of $l^f_2$*, Topology Appl.**34**(1990), no. 2, 153–160. MR**1041769**, DOI 10.1016/0166-8641(90)90077-F - 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. MR**1354937**, DOI 10.4064/fm-148-1-35-62 - Ryszard Engelking,
*Topologia ogólna*, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR**0500779** - D. H. Fremlin,
*Families of compact sets and Tukey’s ordering*, Atti Sem. Mat. Fis. Univ. Modena**39**(1991), no. 1, 29–50. MR**1111757** - Zdeněk Frolík,
*Generalizations of the $G_{\delta }$-property of complete metric spaces*, Czechoslovak Math. J.**10(85)**(1960), 359–379 (English, with Russian summary). MR**116305** - —,
*Topologically complete spaces*, Comment. Math. Univ. Carol.**1**(1960), 1–3. - 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. MR**0145471** - Zdeněk Frolík,
*A survey of separable descriptive theory of sets and spaces*, Czechoslovak Math. J.**20(95)**(1970), 406–467. MR**266757** - Sergei Gul’ko,
*The space $C_p(X)$ for countable infinite compact $X$ is uniformly homeomorphic to $c_0$*, Bull. Polish Acad. Sci. Math.**36**(1988), no. 5-6, 391–396 (1989) (English, with Russian summary). MR**1101684** - S. P. Gul′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. MR**1265990** - R. W. Hansell,
*Descriptive topology*, Recent progress in general topology, M. Hušek and J. van Mill, editors, North-Holland, Amsterdam, 1992, pp. 275–315. - J. R. Isbell,
*Uniform spaces*, Mathematical Surveys, No. 12, American Mathematical Society, Providence, R.I., 1964. MR**0170323** - H. J. K. Junnila and H. P. A. Künzi,
*Characterizations of absolute $F_{\sigma \delta }$-sets*, preprint. - Alexander S. Kechris,
*Classical descriptive set theory*, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR**1321597**, DOI 10.1007/978-1-4612-4190-4 - Victor Klee,
*On the Borelian and projective types of linear subspaces*, Math. Scand.**6**(1958), 189–199. MR**105005**, DOI 10.7146/math.scand.a-10543 - 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** - O. G. Okunev,
*Weak topology of a dual space and a $t$-equivalence relation*, Mat. Zametki**46**(1989), no. 1, 53–59, 123 (Russian); English transl., Math. Notes**46**(1989), no. 1-2, 534–538 (1990). MR**1019256**, DOI 10.1007/BF01159103 - Jean Saint-Raymond,
*La structure borélienne d’Effros est-elle standard?*, Fund. Math.**100**(1978), no. 3, 201–210 (French). MR**509546**, DOI 10.4064/fm-100-3-201-210 - W. Sierpiński,
*Sur une définition topologique des ensembles $F_{\sigma \delta }$*, Fund. Math.**6**(1924), 24–29. - 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). MR**667997** - V. Valov,
*Linear mappings between function spaces*, preprint.

## Additional Information

**Witold Marciszewski**- Affiliation: Vrije Universiteit, Faculty of Mathematics and Computer Science, De Boelelaan 1081 a, 1081 HV Amsterdam, The Netherlands
- Address at time of publication: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
- MR Author ID: 119645
- Email: wmarcisz@cs.vu.nl
**Jan Pelant**- Affiliation: Mathematical Institute of the Czech Academy of Sciences, Žitná 25, 11567 Praha 1, Czech Republic
- Email: pelant@mbox.cesnet.cz
- Received by editor(s): December 14, 1995
- Additional Notes: The first author was supported in part by KBN grant 2 P301 024 07.

The second author was supported in part by the grant GAČR 201/94/0069 and the grant of the Czech Acad. Sci. 119401. - © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**349**(1997), 3585-3596 - MSC (1991): Primary 04A15, 54H05, 54C35
- DOI: https://doi.org/10.1090/S0002-9947-97-01852-7
- MathSciNet review: 1401778