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Transactions of the American Mathematical Society

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

 

 

On topological classification of function spaces $ C\sb p(X)$ of low Borel complexity


Authors: T. Dobrowolski, W. Marciszewski and J. Mogilski
Journal: Trans. Amer. Math. Soc. 328 (1991), 307-324
MSC: Primary 54C35; Secondary 57N17, 57N20
DOI: https://doi.org/10.1090/S0002-9947-1991-1065602-X
MathSciNet review: 1065602
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Abstract: We prove that if $ X$ is a countable nondiscrete completely regular space such that the function space $ {C_p}(X)$ is an absolute $ {F_{\sigma \,\delta }}$-set, then $ {C_p}(X)$ is homeomorphic to $ {\sigma ^\infty }$, where $ \sigma = \{ ({x_i}) \in {{\mathbf{R}}^\infty }:{x_i}= 0$ for all but finitely many $ i\} $. As an application we answer in the negative some problems of A. V. Arhangel'skii by giving examples of countable completely regular spaces $ X$ and $ Y$ such that $ X$ fails to be a $ {b_R}$-space and a $ k$-space (and hence $ X$ is not a $ {k_\omega }$-space and not a sequential space) and $ Y$ fails to be an $ {\aleph _0}$-space while the function spaces $ {C_p}(X)$ and $ {C_p}(Y)$ are homeomorphic to $ {C_p}(\mathfrak{X})$ for the compact metric space $ \mathfrak{X}= \{ 0\} \cup \{ {n^{ - 1}}:n= 1,2, \ldots \} $.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1065602-X
Keywords: Function space, pointwise convergence topology, Borelian filter, $ k$-space, $ {k_\omega }$-space, $ {b_R}$-space, $ {\aleph _0}$-space
Article copyright: © Copyright 1991 American Mathematical Society