Point networks for special subspaces of \(\mathbb {R}^{\kappa}\) (Q2833662)

Three notions of compact subspaces (Eberlein, Talagrad, and Gul'ko) of a Banach space \(X\) with the weak topology are characterized using topological tools of a directed set \(P\), together with order theory, and a collection of subsets (base, almost sub-base, network, or point network) organized by \(P\). A collection \(\mathcal{C} = \bigcup \{ \mathcal{C}_p : \mathcal{C}_p \subseteq X , p \in P \} \) is \(P\)-point finite if (i) \(p \leq p' \) implies that \(\mathcal{C}_p \subseteq \mathcal{C}_{p'} \), and (ii) each \(\mathcal{C}_p\) is point finite. For example, using \(P = \mathbb{N}\), the natural numbers, to characterize Eberlein compactness, the authors define \(X\) to be Eberlein compact if it is compact and \((\mathbb{N} \times \mathbb{N})\)-metacompact, i.e., every open cover of \(X^2 \setminus \Delta\) has an \( (\mathbb{N} \times \mathbb{N})\)-point finite open refinement. A family of sets is \(\mathbb{N}\)-additively Noetherian if every subcollection of the collection of all unions of members of the family, well-ordered by \(\subseteq\), has cardinality less than \( \mathbb{N}^+\). A point network of \(X\) is a collection \(\mathcal{W} = \{\mathcal{W}(x) : x \in X\}\) where each \(\mathcal{W}(x)\) is a collection of subsets of \(X\) such that if \( x \in U\), \(U\) open, then there is an open \(V\) with \(x \in V \subseteq U\), and for \(y \in V\), there is \(W \in \mathcal{W}(y)\), with \(x \in W \subseteq U\). One of the characterizations is that a space \(X\) is Eberlein compact if and only if \(X\) has an \( \mathbb{N}\)-additively Noetherian point network. Similar results for Talagrad and Gul'ko compacta are obtained by using different partial orders associated with them. The authors state in the introduction that the purpose of the paper is ``to give uniform characterizations of certain special subspaces of products of lines that arise in analysis''.