Expandable network and covering properties for uniform Eberlein compacta (Q2502941)

Following A. V. Arkhangel'skii, a family \({\mathcal N}\) of subsets of a topological space \((X,\tau)\) is called a network for \(X\) if for every \(U\in \tau\) and for every \(x\in U\) there exists an \(N\in{\mathcal N}\) such that \(x \in N\subset U\). Moreover, a family \({\mathcal A}\) of subsets is said to be boundedly point finitely expandable if there exist an \(n<\omega\) and a family \(\{U_A\mid A\in{\mathcal A}\}\) of open subsets of \(X\) such that \(A \subset U_A\) for each \(A\in{\mathcal A}\) and, for each \(x\in X\), \(|\{A\in {\mathcal A}\mid x\in U_A\}|<n\). Theorem 1. A compact Hausdorff space is homeomorphic to a weakly compact subset of a Hilbert space (i.e., a uniform Eberlein compact space) iff it has a network \({\mathcal N}=\cup\{ {\mathcal N}_i\mid i<\omega\}\) such that every \({\mathcal N}_i\) is boundedly point finitely expandable. A family \({\mathcal A}\) of subsets of a topological space \(X\) is said to be boundedly point finite if there exists an \(n< \omega\) such that \(|\{A\in{\mathcal A}\mid x\in{\mathcal A}\}|<n\) for each \(x\in X\), and \(X\) is called \(\sigma\)-boundedly metacompact if every open cover \({\mathcal U}\) of \(X\) has an open refinement \({\mathcal V}=\cup \{{\mathcal V}_i\mid i<\omega\}\) such that every \({\mathcal V}_i\) is boundedly point finite. Theorem 2. For a compact Hausdorff space \(X\) the following are equivalent: (i) \(X\) is a uniform Eberlein compact space; (ii) \(X^2\setminus\Delta\) is \(\sigma\)-boundedly metacompact; (iii) \(X^2\) is hereditarily \(\sigma\)-boundedly metacompact. These interesting results nicely complement similar results of \textit{F. García, L. Oncina}, and \textit{J. Orihuela} [J. Math. Anal. Appl. 297, No. 2, 791--811 (2004; Zbl 1081.54018)].