Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T19:47:26.352Z Has data issue: false hasContentIssue false
  • English
  • Français

On properties of metrizable spaces X preserved by t-equivalence

Part of: Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  26 February 2010

Witold Marciszewski
Witold Marciszewski
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland. E-mail: wmarcisz@mimuw.edu.pl
Get access

Abstract

For a completely regular space X, denote by Cp(X) the space of continuous real valued functions on X, endowed with the pointwise convergence topology. The spaces X and Y are t-equivalent if Cp(X) and Cp(Y) are homeomorphic. It is proved that, for metrizable spaces X, the countable dimensionality is preserved by t-equivalence. It is also shown that this relation preserves absolute Borel classes greater than 2 and all projective classes.

MSC classification

Secondary: 54C35: Function spaces
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 273 - 279
Copyright
Copyright © University College London 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[Arl]Arkhangel'skiĭ, A. V.. Topological Function Spaces, (Kluwer Academic Publishers, Dordrecht, 1992).Google Scholar
[Ar2]Arkhangel'skiĭ, A. V.. Cp-Theory, Recent Progress in General Topology (Elsevier, 1992), pp. 1–56.Google Scholar
[Ar3]Arkhangel'skiĭ, A. V.. Embeddings in Cp-spaces. Topology Appl., 85 (1998), 9–33.Google Scholar
[BdGP]Baars, J., Groot, J. de and Pelant, J.. Function spaces of completely mettrizable spaces. Trans. Amer. Math. Soc., 340 (1993), 871–879.CrossRefGoogle Scholar
[Cal]Cauty, R.. L'espace des fonctions continues d'un espace métrique dénombrable. Proc. Amer. Math. Soc., 113 (1991), 493–501.Google Scholar
[Ca2]Cauty, R.. Sur L'invariance de la dimension infinie forte par t-équivalence. Fund. Math., 160 (1999), 95–100.Google Scholar
[DGM]Dobrowolski, T., Gul'ko, S. P. and Mogilski, J.. Function spaces homeomorphic to the countable product of l2f. Topology Appl., 34 (1990), 153–160.Google Scholar
[DM]Dobrowolski, T. and Marciszewski, W.. Classification of function spaces with the pointwise topology determined by a countable dense set. Fund. Math., 148 (1995), 35–62.Google Scholar
[En]Engelking, R.. Theory of Dimension Finite and Infinite, (Helderman Verlag, Lemgo, 1995).Google Scholar
[Gu]Gul'ko, S. P.. On uniform homeomorphisms of spaces of continuous functions. Proc. Steklov lnst. Math., 3 (1992), 87–93.Google Scholar
[GKh]Gul'ko, S. P. and Khmyleva, T. E.. Compactness is not preserved by the relation of t-equivalence. Math. Notes, 39 (1986), 484–488.Google Scholar
[Ke]Kechris, A. S.. Classical Descriptive Set Theory (Springer-Verlag, New York, 1995).CrossRefGoogle Scholar
[Ku]Kuratowski, K.. Topology I. (Academic Press and PWN, New York and London, 1966).Google Scholar
[Ma]Marciszewski, W.. Some recent results on function spaces CP(X). Recent Progress in Function Spaces (Quaderni di Matematica, 3) (ed. Maio, G. Di and Holá, L.) (1998), 221–239.Google Scholar
[MP]Marciszewski, W. and Pelant, J.. Absolute Borel sets and function spaces. Trans. Amer. Math. Soc., 349 (1997), 3585–3596.Google Scholar
[O]Okunev, O. G.. Weak topology of a dual space and a t-equivalence relation. Math. Notes, 46 (1989), 534–538.Google Scholar
[Pe]Pestov, V. G.. The coincidence of the dimension dim of f-equivalent topological spaces. Soviet Math. Dokl., 26 (1982), 380–383.Google Scholar
[Us]Uspenskiĭ, V. V.. A characterization of compactness in terms of uniform structure in a function space. Uspekhi Matem. Nauk, 37 (1982), 183–184.Google Scholar