Disconnectedness of the reduced ideal-product
Given a set \(I\), an ideal \(\Lambda\) on \(I\), a filter \(\Psi\) on \(I\), and an \(I\)-indexed family of topological spaces, the associated reduced ideal product is a topological quotient of a generalized box product. The box topology on the Cartesian product uses sets in \(\Lambda\) to restrict open boxes; the quotient is formed by identifying two \(I\)-sequences if they agree almost everywhere modulo \(\Psi\). This construction generalizes both the box products of \textit{C. J. Knight} [Quart. J. Math. Oxford II. Ser. 15, 41-54 (1964; Zbl 0122.174)] and the topological ultraproducts of the reviewer [Topology Appl. 7, 283-308 (1977; Zbl 0364.54005)]. The author defines the pair \(\langle\Lambda,\Psi\rangle\) to satisfy the \(\Lambda\Psi\)-condition if whenever \(A\in\Psi\) and \(B\not\in\Psi\), there is some \(L\in\Lambda\) with \(I\backslash L\not\in\Psi\) and \(L\subseteq A\backslash B\). This condition is satisfied when \(\Lambda\) extends the ideal of finite sets and \(\Psi=\{I\}\) (box products, including the Tychonoff product and the full box product), as well as when \(\Lambda\) is improper and \(\Psi\) is an ultrafilter (topological ultraproducts). In [\textit{M. Z. Grulović} and \textit{M. S. Kurilić}, ``On preservation of separation axioms in products'' (to appear)], the authors prove that it is precisely for ideal-filter pairs satisfying the \(\Lambda\Psi\)- condition that there is preservation of the separation axioms \(T_ k\), \(k=0,1,2,3,3{1\over 2}\). In the present paper the \(\Lambda\Psi\)-condition is assumed for the main results, which are: (1) reduced ideal products preserve the properties of having trivial quasicomponents and of being zero-dimensional; and (2) when the ideal is countably complete and the filter is \(\omega\)-regular, reduced ideal products of \(T_ 3\) spaces using that ideal-filter pair are strongly zero-dimensional.
scientific article; zbMATH DE number 96610