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On topological classification of function spaces $$C_ p(X)$$ of low Borel complexity. (English) Zbl 0768.54016
For a space $$X$$ define $$C_ p(X)$$ to be the set of all continuous real- valued functions on $$X$$ endowed with the topology of pointwise convergence. The subspace of $$C_ p(X)$$ consisting of all bounded functions is denoted by $$C^*_ p(X)$$. This paper is devoted to the topological classification of $$C_ p(X)$$ and $$C^*_ p(X)$$ for countable completely regular spaces $$X$$. Previously it was proved that for every countable metrizable nondiscrete space $$X$$ the spaces $$C_ p(X)$$ and $$C^*_ p(X)$$ are homeomorphic to $$\sigma^ \infty$$, where $$\sigma=\{(x_ i)\in R^ \infty: x_ i=0$$ for all but finitely many $$i\}$$ [the first and the third author with S. P. Gulko, Topology Appl. 34, No. 2, 153-160 (1990; Zbl 0691.57009)]. Extending this study in this paper the authors focus on the case when $$C_ p(X)$$ is an absolute Borel set. The main result of this paper is the following
Theorem: Let $$X$$ be a countable nondiscrete completely regular space such that the function space $$C_ p(X)$$ is an absolute $$F_{\sigma\delta}$$- set. Then $$C_ p(X)$$ and $$C^*_ p(X)$$ are homeomorphic to $$\sigma^ \infty$$.
The authors also give examples of special $$F_{\sigma\delta}$$-filters, study sequence spaces of higher Borel countability and formulate the following conjecture. For every countable completely regular space $$X$$ such that $$C_ p(X)\in{\mathcal M}_ \alpha\backslash{\mathcal U}_ \alpha$$, $$C_ p(X)$$ is homeomorphic to $$\Omega_ \alpha$$, where for a countable ordinal $$\alpha$$, $${\mathcal M}_ \alpha$$ and $${\mathcal U}_ \alpha$$ denote the class of all absolute Borel sets of the multiplicative and additive class $$\alpha$$, respectively. Three problems are also enunciated.
Reviewer: I.Pop (Iaşi)

MSC:
 54C35 Function spaces in general topology 57N17 Topology of topological vector spaces 57N20 Topology of infinite-dimensional manifolds
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