A characterization of paracompactness of locally Lindelöf spaces (Q1327638)

A space \(X\) is said to have property \({\mathcal B}\) if every infinite open cover \({\mathcal U}\) of X has an open refinement \({\mathcal V}\) such that every point \(x\in X\) has a neighborhood \(W\) such that \(| \{V \in {\mathcal V} : V \cap W \neq \emptyset\} | < | {\mathcal U} |.\) Generally, property \({\mathcal B}\) does not imply paracompactness. The main result of the paper states that a locally Lindelöf space with property \({\mathcal B}\) is paracompact. A. V. Arkhangelskij proved in the early seventies that the locally compact perfectly normal metacompact spaces are compact. It is proved in this paper that `locally compact' can be weakened to `locally Lindelöf' in this theorem and a question is rised whether `metacompact' can be replaced by `submetacompact'. Another problem stated in the paper is the following: does every locally compact normal metacompact space have property \({\mathcal B}\)?