On the \(M_3\) versus \(M_1\) problem (Q1570912)

The authors prove two significant results: Theorem 15. If \(X\) is an \(M_3\)-space with property \((P)\) then \(X\) is an \(M_1\)-space such that every closed subset of \(X\) has a CP open neighborhood base in \(X\). CP means closure-preserving; property \((P)\): Every open \(U\subset X\) satisfies the following: \(p\in \partial U\) implies that there is a CP family \({\mathcal G}\) of closed subsets of \(U^-\) such that, for each \(G\in{\mathcal G}\), \((G\cap U)^-=G\), and \(p \in\) open \(O\) implies \(p\in G\subset O\) for some \(G\in {\mathcal G}\). Theorem 16. An \(M_3\)-space which is also a \(k\)-space has property \((P)\).