A characterization of monotonically Lindelöf generalized ordered spaces

A space X is said to be monotonically Lindelöf if for any open cover U of X, there exists a countable open cover rU such that rU refines U and if an open cover V of X refines U, then rV refines rU. Gruenhage shows that every Lindelöf first-countable GO-space is monotonically Lindelöf. In the paper we obtain the following result:A Lindelöf GO-space X is monotonically Lindelöf if and only if(1)X has no subspace homeomorphic to a stationary subset of a regular ordinal ⩾ω1;(2)the closure of the set of all points with character ω1 is monotonically Lindelöf.