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1993 | 142 | 3 | 269-301
Tytuł artykułu

A contribution to the topological classification of the spaces Ср(X)

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Abstrakty
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We prove that for each countably infinite, regular space X such that $C_p(X)$ is a $Z_σ$-space, the topology of $C_p(X)$ is determined by the class $F_0(C_p(X))$ of spaces embeddable onto closed subsets of $C_p(X)$. We show that $C_p(X)$, whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set $Ω_α$ for the multiplicative Borel class $M_α$ if $F_0(C_p(X)) = M_α$. For each ordinal α ≥ 2, we provide an example $X_α$ such that $C_p(X_α)$ is homeomorphic to $Ω_α$.
Twórcy
autor
  • Université Paris VI, Analyse Complexe et Géométrie, 4, Place Jussieu, 75252 Paris Cedex 05, France
  • Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland, marcisz@mimuw.edu.pl
Bibliografia
  • [1] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa 1975.
  • [2] M. Bestvina and J. Mogilski, Characterizing certain incomplete infinite-dimensional retracts, Michigan Math. J. 33 (1986), 291-313.
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  • [16] T. Dobrowolski and H. Toruńczyk, Separable complete ANR's admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), 229-235.
  • [17] R. Engelking, General Topology, PWN, Warszawa 1977.
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  • [20] K. Kuratowski, Topology. I, Academic Press, New York 1966.
  • [21] L. A. Louveau and J. Saint Raymond, Borel classes and games: Wadge-type and Hurewicz-type results, Trans. Amer. Math. Soc. 304 (1987), 431-467.
  • [22] D. Lutzer, J. van Mill and R. Pol, Descriptive complexity of function spaces, ibid. 291 (1985), 121-128.
  • [23] W. Marciszewski, On analytic and coanalytic function spaces $C_p(X)$, Topology Appl., to appear.
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  • [25] H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of $l_2$-manifolds, ibid. 101 (1978), 93-110.
  • [26] W. W. Wadge, Ph.D. thesis, Berkeley 1984.
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