The \(\ell_1\)-dichotomy theorem with respect to a coideal
An ideal \(\mathcal{I}\) on \(\mathbb{N}\) is a family of subsets of \(\mathbb{N}\) closed under taking finite unions and subsets of its elements. Assume, moreover, that \(\mathcal{I}\) is proper (\(\mathbb{N}\notin\mathcal{I}\)) and contains all finite sets. If \(\mathcal{I}\) is an ideal, then the family \(\mathcal{P}\setminus\mathcal{I}\) is called a coideal. A family \(\mathcal{B}\) is a coideal basis on \(\mathbb{N}\) if the family \(\mathcal{L}_\mathcal{B}=\{ A\subset \mathbb{N}: \text{ there exists }B\in\mathcal{B} \text{ with } B\subset A\}\) is a coideal. The paper under review consists of an introduction and three sections. In the first section, the authors define and examine three properties of a coideal basis: Ramsey-likeness, selectivity and semiselectivity. For example, a coideal basis \(\mathcal{B}\) is Ramsey if for every \(n,r\in\mathbb{N}\) and for every \(A\in\mathcal{L}_\mathcal{B}\) with \([A]^n=C_1\cup\ldots\cup C_r\), there exist \(B\in\mathcal{B}\), \(B\subset A\) and \(1\leq i \leq r\) such that \([B]^n\subset C_i\). In the next section, the authors introduce, for a given coideal basis \(\mathcal{B}\), the notions of a \(\mathcal{B}\)-sequence, a \(\mathcal{B}\)-subsequence and a \(\mathcal{B}\)-convergent sequence in a metric space. A \(\mathcal{B}\)-sequence in \(X\) is a function \(a:A\to X\), where \(A\in\mathcal{L}_\mathcal{B}\). A \(\mathcal{B}\)-sequence \(\{ x_n\}_{n\in B}\) is called a \(\mathcal{B}\)-subsequence of \(\{ x_n\}_{n\in A}\) if \(B\subset A\). A \(\mathcal{B}\)-sequence \(\{ x_n\}_{n\in B}\) \(\mathcal{B}\)-converges to \(x\in X\) if for every \(\varepsilon>0\) the set \(\{ n\in B: \varrho(x_n,x)\geq\varepsilon\}\) is small, i.e., does not belong to \(\mathcal{L}_\mathcal{B}\). The following version of the Bolzano-Weierstrass theorem is proven: If \(\mathcal{B}\) is a Ramsey coideal basis, then every bounded \(\mathcal{B}\)-sequence of real numbers has a \(\mathcal{B}\)-convergent subsequence. In the last section, the authors establish the following version of the fundamental Rosenthal \(\ell_1\)-dichotomy theorem: If \(\mathcal{B}\) is a semiselective coideal basis, then every bounded \(\mathcal{B}\)-sequence of real-valued functions defined on an infinite set \(X\), \(\{ f_n\}_{n\in A}\), has a \(\mathcal{B}\)-subsequence \(\{ f_n\}_{n\in B}\) which is either \(\mathcal{B}\)-convergent or equivalent to the unit vector basis of \(\ell_1(B)\). Reviewer's remark: It seems that many of the results from Section 2 have been previously reported by \textit{R. Filipów} et al. in the language of ideal convergence, see [J. Symb. Log. 72, No. 2, 501--512 (2007; Zbl 1123.40002); Czech. Math. J. 61, No. 2, 289--308 (2011; Zbl 1249.05378)], or [J. Math. Anal. Appl. 396, No. 2, 680--688 (2012; Zbl 1258.40001)].
scientific article; zbMATH DE number 6848945