A first countable linearly Lindelöf not Lindelöf space
by
Oleg Pavlov
University of North Carolina at Charlotte

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) < ℵ_w.

Arhangel'skii and Buzyakova proved that the cardinality of a first countable linearly Lindelöf space does not exceed 2^ℵ₀. Consequently, a first countable linearly Lindelöf space is Lindelöf if ℵ_w > 2^ℵ₀. 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 ℵ_w. We answer this question in the negative by constructing a counterexample from MA+ℵ_w < 2^ℵ₀.

Date received: June 16, 2006