Characterizations of expandability and the \({\mathcal B}\)-property (Q810918)

Let \(\kappa\) and \(\lambda\) be infinite cardinals with \(\lambda\leq \kappa\) and I(\(\kappa\),\(\lambda\)) the topological space whose underlying set is \(\kappa +1\) and whose topology is given by the collection \(\{U| \quad U\subset \kappa \}\cup \{U| \quad k\in U\subset k+1,\quad | \kappa -U| <\lambda \}.\) The author defines (\(\kappa\),\(\lambda\))- expandability and (\(\kappa\),\(\lambda\))-paracompactness of a regular \(T_ 1\)-space X which are general versions of expandability and paracompactness, respectively, and proves mainly the following theorems: (1) X is (\(\kappa\),\(\lambda\))-expandable (that is, for every locally- \(\lambda\kappa\)-sequence \(\{F_{\alpha}|\alpha <\kappa \}\) of closed sets of X, there is a locally-\(\lambda\kappa\)-sequence \(\{U_{\alpha}|\alpha <\kappa \}\) of open sets of X such that \(F_{\alpha}\subset U_{\alpha}\) for every \(\alpha <\kappa)\) iff any closed set F and \(X\times \{\kappa \}\) in \(X\times I\{\kappa,\lambda \}\) with \(F\cap (X\times \{\kappa \})=\emptyset\) are separated by disjoint open sets. (2) If X has the strong \({\mathcal B}(\mu)\)-property (every open cover of size \(\leq \mu\) has a star-\(\mu\) open refinement) for every \(\mu\leq \kappa\), then X is (\(\kappa\),\(\omega\))-paracompact (that is, every open cover of size \(\leq \kappa\) has a locally-\(\omega\) \((=\) locally finite) open refinement.