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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: [email protected]
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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

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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