Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Condensations of projective sets onto compacta

Author: Henryk Michalewski
Journal: Proc. Amer. Math. Soc. 131 (2003), 3601-3606
MSC (2000): Primary 54C35, 03E15, 28A05, 54H05
Published electronically: April 1, 2003
MathSciNet review: 1991774
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a coanalytic-complete or ${\vtop{\hbox{${\boldsymbol{\Pi}}^{1}_{2}$ } \kern-4pt \hbox{\kern4pt$\widetilde{}$ }\kern-6pt}} $-complete subspace $X$ of a Polish space we prove that there exists a continuous bijection of $X$ onto the Hilbert cube $[0,1]^{\mathbb{N} }$. This extends results of Pytkeev. As an application of our main theorem we give an answer to some questions of Arkhangelskii and Christensen.

Under the assumption of Projective Determinacy we also give some generalizations of these results to higher projective classes.

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

  • 1. A. Andretta, A. Marcone, Pointwise convergence and the Wadge hierarchy, Comment. Math. Univ. Carolinae 42 (2001), 159-172. MR 2002c:03073
  • 2. A.V. Arkhangelskii, On condensations of $C_p$-spaces onto compacta, Proc. Amer. Math. Soc. 128 (2000), 1881-1883. MR 2001c:54002
  • 3. J.P.R. Christensen, Topology and Borel Structure, North-Holand, Amsterdam, 1974. MR 50:1221
  • 4. A.S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. MR 96e:03057
  • 5. W. Kulpa, On a problem of Banach, Colloquium Math. 56 (1988), 255-261. MR 91d:54023
  • 6. K. Kuratowski, Topology, vol. I, New York, 1966. MR 36:840
  • 7. A. Louveau, J. Saint-Raymond, Borel classes and closed games: Wadge-type results and Hurewicz-type results, Trans. Amer. Math. Soc. 304 (1987), 431-467. MR 89g:03068
  • 8. E.G. Pytkeev, The upper bounds of topologies (in Russian), Mat. Zametki 20 no. 4 (1976), 489-500. MR 55:1262
  • 9. The Scottish Book, ed. R.D. Mauldin, Birkhauser, Boston, 1981. MR 84m:00015

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54C35, 03E15, 28A05, 54H05

Retrieve articles in all journals with MSC (2000): 54C35, 03E15, 28A05, 54H05

Additional Information

Henryk Michalewski
Affiliation: Institute of Mathematics, Warsaw University, S. Banach 2 st., 02–097 Warsaw, Poland

Keywords: Function spaces, topology of pointwise convergence, condensation, coanalytic set, projective set, uniformization
Received by editor(s): December 12, 2001
Received by editor(s) in revised form: May 21, 2002
Published electronically: April 1, 2003
Additional Notes: The author’s research was partially supported by KBN Grant 5PO3A02321
Communicated by: Alan Dow
Article copyright: © Copyright 2003 American Mathematical Society