Galois connections between generating systems of sets and sequences
For a Banach space \(X\), consider the collection \(\mathbf{K}\) of all compact sets of \(X\) and the collection \(\mathbf{c}\) of all convergent sequences of \(X\). Obviously, we may characterize \(\mathbf{K}\) in terms of \(\mathbf{c}\): A set \(K\) is in \(\mathbf{K}\) if every sequence of \(K\) has a subsequence which belongs to \(\mathbf{c}\). This way to generate a collection of sets from a collection of sequences was first studied by [\textit{I. Stephani}, Math. Nachr. 99, 13--27 (1980; Zbl 0474.47019)], where she introduced the notions of generating systems of sets (\(\mathrm{GSet}\)) and generating systems of sequences (\(\mathrm{GSeq}\)). This construction gives an operator \(\Psi: \mathrm{GSeq}\rightarrow \mathrm{GSet}\). In the article under review, the author takes up Stephani's work and defines the operator \(\Phi: \mathrm{GSet}\rightarrow \mathrm{GSeq}\) as follows: for \(\mathbf{G} \in \mathrm{GSet}\), a sequence belongs to \(\Phi(\mathbf{G})\) if and only if it is contained in some \(G \in \mathbf{G}\). Since \(\mathrm{GSet}\) is ordered by the inclusion, Stephani defined in \(\mathrm{GSeq}\) a preorder. The author shows that this preorder leads to an equivalence relation \(\sim\) on \(\mathrm{GSeq}\) and induces an order on \(\mathrm{GSeq}/{\sim}\). The saturated systems of sequences, \(\mathrm{satGSeq}\), of all collection of sequences \(\mathbf g\) is considered such that \(\mathbf{g}=\Phi \circ \Psi(\mathbf{g})\) and the sequentially generatable systems of sets, \(\mathrm{seqGSet}\), as all \(\mathbf{G} \in \mathrm{GSet}\) such that \(\mathbf{G}=\Phi(\mathbf{g})\) for some \(\mathbf{g} \in \mathrm{GSeq}\). Then the operators \(\phi: \mathrm{GSet}\rightarrow \mathrm{GSeq}/{\sim}\) and \(\psi: \mathrm{GSeq}/{\sim}\rightarrow \mathrm{GSet}\) are defined as \[ \phi(\mathbf{G})=[\Phi(\mathbf{G}] \quad \text{and} \quad \psi([\mathbf{g}])=\Psi(\mathbf{g}), \] and it is shown that the pair \((\phi,\psi)\) is a Galois connection between the ordered sets GSet and \(\mathrm{GSeq}/{\sim}\) (Theorem 4.6). This result is used, among other things, to show that {\parindent=0.7cm \begin{itemize}\item[{\(\bullet\)}] \(\psi: \mathrm{GSeq}/{\sim} \rightarrow \mathrm{seqGset}\) is an order-isomorphism, \item[{\(\bullet\)}] \(\Phi: \mathrm{seqGset}\rightarrow \mathrm{satGSeq}\) is an order-isomorphism, \item[{\(\bullet\)}] \(\Phi \circ \psi: \mathrm{GSeq}/{\sim}\rightarrow \mathrm{satGSeq}\) is an order-isomorphism. \end{itemize}} In the last section, it is shown that \(\mathrm{GSet}\), \(\mathrm{seqGSet}\), \(\mathrm{satGSeq}\) and \(\mathrm{Gseq}/{\sim}\) are complete lattices.
scientific article; zbMATH DE number 6816285
scientific article