A topological space X is called linearly Lindel&#
xf6;f if every increasing open cover of X has a countable subcover. It is well
known that every Lindel&#
xf6;f space is linearly Lindel&#
xf6;f. The converse implication holds only in particular cases, such as X being countably paracompact or if nw(X)<
ω.
Arhangelskii and Buzyakova proved that the cardinality of a first countable linearly Lindelöf space does not exceed 20. Consequently, a first countable linearly Lindelöf space is Lindelöf if ω>20. 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 ω. We answer this question in the negative by constructing a counterexample from MA+ω<20.
A modification of Alsters Michael space that is first countable is presented.
