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On a typical property of functions. (English) Zbl 0835.26005
Let \(s\) be the space of all real sequences endowed with the Fréchet metric \(\rho\), \[ \rho(a, b)= \sum^\infty_{k= 1} 2^{- k}|a_k- b_k|/(1+ |a_k- b_k|),\;a= \{a_k\}_k,\;b= \{b_k\}_k\in s. \] Let \(\mathcal F\) denote the space of all functions \(f: \mathbb{R}\to \mathbb{R}\) with the metric of uniform convergence. A set \(E\subset s\) is said to be superporous at \(a\in s\), if \(E\cup F\) is porous at \(a\) whenever \(F\subset s\) is porous at \(a\). The set \(E\) is said to be superporous if it is superporous at each of its points and \(E\) is said to be \(\sigma\)-superporous if it is a countable union of superporous sets.
The authors deal with the class \(\mathcal U\) of all functions \(f\in {\mathcal F}\) for which the set {\(\{a_k\}_k\in s: \sum_k f(a_k)\) converges} is \(\sigma\)-superporous in \((s, \rho)\). They show that \(\mathcal U\) is residual in \(\mathcal F\), both \(\mathcal U\) and \({\mathcal F}- {\mathcal U}\) are dense-in-itself and \(\mathcal U\) is a Baire space in the relative topology.

MSC:
26A21 Classification of real functions; Baire classification of sets and functions
40A05 Convergence and divergence of series and sequences
54E52 Baire category, Baire spaces
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