The monotone Lindelöf property and separability in ordered spaces (Q555811)

A topological space \(X\) is called monotonically Lindelöf if for each open cover \({\mathcal U}\) of \(X\) there is a countable open cover \(r({\mathcal U})\) of \(X\) that refines \({\mathcal U}\) and has the property that if an open cover \({\mathcal U}\) refines an open cover \({\mathcal V},\) then \(r({\mathcal U})\) refines \(r({\mathcal V}).\) The authors investigate the monotone Lindelöf property in generalized ordered spaces (= GO-spaces). They present a monotonically Lindelöf LOTS that is not first countable. They also observe that a compact monotonically Lindelöf LOTS is first countable and that the lexicographic square \([0,1]\times [0,1]\) is compact and monotonically Lindelöf, but not perfect. Moreover they prove that any separable GO-space is hereditarily monotonically Lindelöf. Given any \(M\) model of ZFC, they state, that if \(M\) contains a Souslin line, then \(M\) has a GO-space that is hereditarily monotonically Lindelöf but not separable; and if \(M\) does not contain a Souslin line, then separability is equivalent to the hereditary monotone Lindelöf property for any GO-space in \(M.\)