A first countable linearly Lindelöf not Lindelöf topological space

A topological space X is called linearly Lindelöf if every increasing open cover of X has a countable subcover. It is well known that every Lindelöf space is linearly Lindelöf. The converse implication holds only in particular cases, such as X being countably paracompact or if nw(X)<ℵω.Arhangelʼskii and Buzyakova proved that the cardinality of a first countable linearly Lindelöf space does not exceed ℵ02. Consequently, a first countable linearly Lindelöf space is Lindelöf if ℵω>ℵ02. They asked whether every linearly Lindelöf first countable space is Lindelöf in ZFC. This question is supported by the fact that all known linearly Lindelöf not Lindelöf spaces are of character at least ℵω. We answer this question in the negative by constructing a counterexample from MA+ℵω<ℵ02.A modification of Alsterʼs Michael space that is first countable is presented.