More On Cardinality Bounds Involving the Weak Lindelof Degree

Abstract

We give several new bounds for the cardinality of a Hausdorff topological space X involving the weak Lindel¨of degree ^wL(X). In particular, we show that if X is extremally disconnected, then |X| ≤ 2 ^wL(X)^πχ(X)^ψ(X) , and if X is additionally power homogeneous, then |X| ≤ 2 ^wL(X)^πχ(X) . We also prove that if  X is a star-DCCC space with a Gδ-diagonal of rank 3, then |X| ≤ 2 ℵ0 ; and if X is any normal star-DCCC space with a Gδ-diagonal of rank 2, then |X| ≤ 2 ℵ0 . Several improvements of results in [10] are also given. We show that if X is locally compact, then |X| ≤ ^wL(X) ψ(X) and that |X| ≤ ^wL(X) t(X) if X is  additionally power homogeneous. We also prove that |X| ≤ 2 ψc(X)t(X)^wL(X) for any space with a π-base whose elements have compact closures and that  the stronger inequality |X| ≤ ^wL(X) ψc(X)t(X) is true when X is locally H-closed or locally Lindelof.