A note on star Lindelöf, first countable and normal spaces

Abstract

A topological space X is said to be star Lindelöf if for any open cover U of X there is a Lindelöf subspace A ⊂ X such that St(A, 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 MA + ¬CH, which shows that a star Lindelöf, first countable and normal space may not have countable extent.