A note on star Lindelöf, first countable and normal spaces

summary:A topological space $X$ is said to be star Lindelöf if for any open cover $\mathcal U$ of $X$ there is a Lindelöf subspace $A \subset X$ such that $\operatorname {St}(A, \mathcal U)=X$. The “extent” $e(X)$ of $X$ is the supremum of the cardinalities of closed discrete subsets of $X$. We prove that under $V=L$ every star Lindelöf, first countable and normal space must have countable extent. We also obtain an example under $\rm MA +\nobreak \neg CH$, which shows that a star Lindelöf, first countable and normal space may not have countable extent