Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Absolute Borel sets and function spaces

Authors: Witold Marciszewski and Jan Pelant
Journal: Trans. Amer. Math. Soc. 349 (1997), 3585-3596
MSC (1991): Primary 04A15, 54H05, 54C35
MathSciNet review: 1401778
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 Cech-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 \v{C}ech-completeness of a metric space $X$ is determined by the linear structure of $C_{p}(X)$.

References [Enhancements On Off] (What's this?)

  • [A1] A. V. Arkhangel'skii, Topological Function Spaces, Kluwer Academic Publishers, Dordrecht, 1992. MR 92i:54022
  • [A2] -, On topological spaces which are complete in the sense of Cech, Vest. Mosk. Univ. (1961), 37-40.MR 24:A1110
  • [BGP] J. Baars, J. de Groot, J. Pelant, Function spaces of completely metrizable spaces, Trans. Amer. Math. Soc. 340 (1993), 871-879. MR 94f:54036
  • [Ba] A. Barbati, The hyperspace of an analytic metrizable space is analytic, Proc. 11th Int. Conf. on Topology (Trieste 1993); Rend. Ist.Mat. Trieste 25 (1993), 15-21. MR 96c:54047
  • [Be] G. Beer, Topologies on closed and closed convex sets, Kluwer Academic Publishers, Dordrecht, 1993. MR 95k:49001
  • [BP] C. Bessaga and A. Pelczynski, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa, 1975. MR 57:17657
  • [Ch] J.P.R. Christensen, Topology and Borel Structure, North-Holland Publishing Company, Amsterdam, London, 1974. MR 50:1221
  • [Co] C. Costantini, Every Wijsman topology relative to a Polish space is Polish, Proc. Amer. Math. Soc 123 (1995), 2569-2574. MR 95j:54012
  • [CLP] C. Costantini, S. Levi, J. Pelant, On compactness in hyperspaces, in preparation.
  • [DGM] T. Dobrowolski, S. P. Gulko and J. Mogilski, Function spaces homeomorphic to the countable product of $\ell ^{2}_{f}$, Topology Appl. 34 (1990), 153-160. MR 91c:57024
  • [DM] T. Dobrowolski and W. Marciszewski, Classification of function spaces with the pointwise topology determined by a countable dense set, Fund. Math. 148 (1995), 35-62. MR 96k:54017
  • [En] R. Engelking, General topology, PWN, Warszawa, 1977. MR 58:18316b
  • [Fr] D. H. Fremlin, Families of compact sets and Tukey's ordering, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), 29-50. MR 92c:54032
  • [F1] Z. Frolík, Generalization of the $G_{\delta }$-property of complete metric spaces, Czech. Math. J. 10 (1960), 359-379. MR 22:7100
  • [F2] -, Topologically complete spaces, Comment. Math. Univ. Carol. 1 (1960), 1-3.
  • [F3] -, A contribution to descriptive theory of sets and spaces, General Topology and its Relations to Modern Analysis and Algebra, J. Novák, ed., Academia, Prague, 1962, pp. 157-173. MR 26:3002
  • [F4] -, A survey of separable descriptive theory of sets and spaces, Czech. Math. J. 20 (1970), 406-467. MR 42:1660
  • [Gu1] S. P. Gul'ko, The space $C_p(X)$ for countable infinite compact $X$ is uniformly homeomorphic to $c_{0}$, Bull. Pol. Acad. Sci. 36 (1988), 391-396. MR 92e:54011
  • [Gu2] -, O ravnomernykh gomeomorfizmakh prostranstv nepreryvnykh funktsii, Baku International Topological Conference (1987), Trudy Mat. Inst. Steklova 193 (1990) (Russian); English transl. in Steklov Inst. Math. (1993), 87-93. MR 95f:54017
  • [Ha] 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
  • [Is] J. R. Isbell, Uniform spaces, Math. Surv. 12, A.M.S., Providence, Rhode Island, 1964. MR 30:561
  • [JK] H. J. K. Junnila and H. P. A. Künzi, Characterizations of absolute $F_{\sigma \delta }$-sets, preprint.
  • [Ke] A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. MR 96e:03057
  • [Kl] V. L. Klee, On the Borelian and projective types of linear subspaces, Math. Scand. 6 (1958), 189-199. MR 21:3752
  • [Ku] K. Kuratowski, Topology I, Academic Press and PWN, New York and London, 1966. MR 36:840
  • [O] O. G. Okunev, O slabotopologii soprazhennogo prostranstva i otnoshenii t-ekvivalentnosti, Mat. Zametki 46 (1989), 53-59 (Russian); English transl., Weak topology of a dual space and a $t$-equivalence relation, Math. Notes 46 (1989), 534-536. MR 91h:46008
  • [SR] J. Saint Raymond, La structure borélienne d'Effros est-elle standard?, Fund. Math. 100 (1978), 201-210. MR 80g:54044
  • [Si] W. Sierpinski, Sur une définition topologique des ensembles $F_{\sigma \delta }$, Fund. Math. 6 (1924), 24-29.
  • [Us] V. V. Uspenskii, A characterization of compactness in terms of uniform structure in a function space, Uspekhi Matem. Nauk 37 (1982), 183-184. MR 83k:54025
  • [Va] V. Valov, Linear mappings between function spaces, preprint.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 04A15, 54H05, 54C35

Retrieve articles in all journals with MSC (1991): 04A15, 54H05, 54C35

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

Jan Pelant
Affiliation: Mathematical Institute of the Czech Academy of Sciences, Žitná 25, 11567 Praha 1, Czech Republic

Keywords: Absolute Borel set, function space
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.
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society