On diagonal degrees and star networks (Q6633959)

Throughout, any separation axioms which are required are specifically mentioned. Given a cover \(\mathcal U\) of a space \(X\) and an integer \(m \geq 1\), a \textit{star-m network} (respectively, a \textit{weak star-m network} for \(\mathcal U\) is a collection \(\mathcal N\) of subsets of \(X\) such that for all \(x \in X\) there exists \(N\in\mathcal N\) such that \(x \in N \subseteq St^m(x,\mathcal U)\) (respectively, \(x \in N \subseteq \mathrm{cl}(St^m(x,\mathcal U))\). Using the concepts of a star network and a weak star network, cardinal functions \(sn_m(X)\) and \(wsn_m(X)\) are defined for each \( m\in \mathbb N\); specifically, \(sn_m(X)\) (respectively, \(wsn_m(X)\)) is the minimal cardinal \(\kappa\) such that every open cover of \(X\) has a star-\(m\) network (respectively, a weak star-\(m\) network). After an introductory section and a section of further definitions, in Section 3, the author uses the cardinal functions \(sn_m(X)\) and \(wsn_m(X)\) and a number of others to obtain cardinal inequalities which improve known results. For example, it is shown that \(sn_1(X)\leq L(X)\) (where as usual, \(L(X)\) denotes the Lindelöf degree of \(X\)) for any space \(X\) and then that \(|X|\leq sn_1(X)^ {\Delta(X)}\) (where \(\Delta(X)\) denotes the diagonal degree of \(X\)), thus improving the inequality \(|X|\leq L(X)^{\Delta(X)}\) of O. T. Alas. Another inequality proved in Section 3, (Theorem 3.17), is that \(sn_3(X)\leq wL(X)\) (the weak Lindelöf number of \(X\)) and this inequality is used to improve an inequality due to \textit{D. Basile} et al. [Houston J. Math. 40, No. 1, 255--266 (2014; Zbl 1293.54003)]. Section 4 contains many results of a similar type concerning the cardinal \(\overline{sn}(X)\), being the least cardinal \(\kappa\) such that every open cover of \(X\) has a regular star network -- that is to say, a collection \(\mathcal N\) of subsets of \(X\) such that for all \(x\in X\), there exists \(N\in\mathcal N\) such that \(x\in N\subseteq \bigcap\{\overline{St(U,\mathcal U)}: x\in U\in\mathcal U\}\) -- of cardinality at most \(\kappa\). Among them, \({\overline{sn}}(X)\leq wL(X)^{\chi(X)}\) and \(\overline{sn}(X)\leq 2^{c(X)},\) from which generalizations of cardinal inequalities of Gotchev and Buzyakova are obtained.