Dual discreteness of \(\Sigma \)-products and irreducibility of infinite products (Q6611771)

All spaces considered in this paper are topological \(T_1\)-spaces, \(\mathbb{N}\) is an infinite countable discrete space, \(\omega\) is the first infinite cardinal, and \(\omega_1\) is the first uncountable cardinal.\N\NWe recall that a \textit{neighborhood assignment} for a space \((X,\tau)\) is a function \(U : X \rightarrow \tau\) such that \(x \in U(x)\) for each \(x \in X\). A space \(X\) is a \(D\)-\textit{space} if for every neighborhood assignment \(U\) for \(X\), there is a closed discrete subset \(D\) in \(X\) such that \(\{U(d) : d \in D\}\) covers \(X\). \(X\) is \textit{dually discrete} if for every neighborhood assignment \(U\) for \(X\), there is a discrete subset \(D\) of \(X\) such that \(\{U(d) : d \in D\}\) covers \(X\). A space \(X\) is an \(aD\)-\textit{space} if for every closed subset \(F\) of \(X\) and every open cover \(\mathcal{U}\) of \(X\), there is a closed discrete subset \(D\) of \(F\) such that for every \(d\in D\) one can assign \(U(d) \in \mathcal{U}\) with \(d\in U(d)\) such that \(\{U(d) : d \in D\}\) covers \(F\). A cover \(\mathcal{U}\) of a set \(S\) is \textit{minimal} if \(\mathcal{U}\setminus\{U\}\) does not cover \(S\) for any \(U \in\mathcal{U}\). A space \(X\) is \textit{irreducible} if every open cover of \(X\) has a minimal open refinement. We recall that every \(D\)-space is dually discrete and is an \(aD\)-space, and that the \(aD\)-spaces are irreducible.\N\NLet \(\left\{X_i:i\in I\right\}\) be a family of non-empty topological spaces and \(p=\{p_i:i\in I\}\) be a point in the product \(X_I\) of these spaces. The \(\Sigma\)-\textit{product} at \(p\), denoted by \(\Sigma(p)\), is the subspace of \(X_I\) that contains all points \(x=\{x_i:i\in I\}\) for which \(|\{i\in I:x_i \neq p_i\}|\le\omega\).\N\NThe authors of the paper under review proved in their previous paper [Topology Appl. 195, 297--311 (2015; Zbl 1329.54025)] that \(\mathbb{N}^{\omega_1}\) is not a \(D\)-space, and in this paper they show the following:\N\N1. Every \(\Sigma\)-product of compact metric spaces is dually discrete.\N\N2. Under \(CH\), \(\mathbb{N}^{\omega_1}\) is not dually discrete.\N\N3. \(\mathbb{N}^{\omega_1}\) is not an \(aD\)-space and therefore, \(\mathbb{R}^{\omega_1}\) is not an \(aD\)-space.\N\N4. \(\mathbb{N}^{\omega_1}\) is irreducible.