On topological classification of function spaces \(C_ p(X)\) of low Borel complexity.

*(English)*Zbl 0768.54016For 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.

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 |