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Transactions of the American Mathematical Society

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



On topological classification of function spaces $C_ p(X)$ 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
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’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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Keywords: Function space, pointwise convergence topology, Borelian filter, <IMG WIDTH="17" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img15.gif" ALT="$k$">-space, <!– MATH ${k_\omega }$ –> <IMG WIDTH="28" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${k_\omega }$">-space, <IMG WIDTH="27" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="images/img16.gif" ALT="${b_R}$">-space, <!– MATH ${\aleph _0}$ –> <IMG WIDTH="27" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="images/img17.gif" ALT="${\aleph _0}$">-space
Article copyright: © Copyright 1991 American Mathematical Society