Some characterizations of a B-property (Q1821379)

A topological space X has the B-property [\textit{P. Zenor}, Proc. Am. Math. Soc. 24, 258-262 (1970; Zbl 0189.533)] if, for any monotone increasing open covering \(\{U_{\alpha}| \alpha <\tau \}\) of X, there exists a monotone increasing open covering \(\{V_{\alpha}| \alpha <\tau \}\) of X such that \(cl(V_{\alpha})\subset U_{\alpha}\) for each \(\alpha <\tau\), where \(cl(V_{\alpha})\) denotes the closure of \(V_{\alpha}\). This property is in between paracompactness and countable paracompactness. The author gives equivalences for this property and shows that a regular space which is para-Lindelöf and countably paracompact satisfies the B-property. The author also discusses some product theorems involving this property.