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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On topological classification of function spaces $C_ p(X)$ of low Borel complexity
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by T. Dobrowolski, W. Marciszewski and J. Mogilski PDF
Trans. Amer. Math. Soc. 328 (1991), 307-324 Request permission

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’skiǐ 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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Additional Information
  • © Copyright 1991 American Mathematical Society
  • 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