Domination in products
All spaces are Tychonoff. For a space \(X\) denote by \(\mathcal K(X)\) the set of all compact subsets of \(X\). A family \(\mathcal A\) of subsets of \(X\) is fundamental if for every \(K\in\mathcal K(X)\) there is \(A\in \mathcal A\) with \(K\subset A\); \(\mathcal A\) is \(M\)-ordered for some space \(M\) if \(\mathcal A=\{A_K:K\in \mathcal K(M)\}\) and \(K\subset L\) implies \(A_K\subset A_L\). A space \(X\) is \(M\)-dominated (strongly \(M\)-dominated) if it has an \(M\)-ordered (\(M\)-ordered fundamental) compact cover. \textit{D. Guerrero Sánchez} [Topology Appl. 160, No. 13, 1652--1658 (2013; Zbl 1286.54005)] defined the following two cardinal invariants for a space \(X\): the metric domination index of \(X\), \(dm(X) = \min\{w(M) : X \text{ is }M\)-dominated by a metric space \(M\)
scientific article; zbMATH DE number 6473611