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.
A topological space X has a rank 2-diagonal if there exists a diagonal sequence on X of rank 2, that is, there is a countable family {Un : n ∈ ω} of open covers of X such that for each x ∈ X, {x} = ∩ {St2 (x,Un): n ∈ ω}. We say that a space X satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of X is countable. We mainly prove that if X is a DCCC normal space with a rank 2-diagonal, then the cardinality of X is at most c. Moreover, we prove that if X is a first countable DCCC normal space and has a Gδ-diagonal, then the cardinality of X is at most c.