More on cardinality bounds involving the weak Lindelöf degree

Abstract

We give several new bounds for the cardinality of a Hausdorff topological space X involving the weak Lindelöf degree ωL(X). In particular, we show that if X is extremally disconnected, then |X| ≤ 2 ^{ωL (X)πχ(X)ψ(X)}, and if X is additionally power homogeneous, then |X| ≤2 ^{ωL (X)πχ(X)}. We also prove that if X is a star-DCCC space with a G_δ -diagonal of rank 3, then |X| ≤ 2^ℵ0 ; and if X is any normal star-DCCC space with a G_δ -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| ≤ ωL(X) ^{ψ (X)} and that |X| ≤ ωL(X) ^{t (X)} if X is additionally power homogeneous. We also prove that |X| ≤ 2 ^{ψc (X)t(X) ωL(X)} for any space with a π-base whose elements have compact closures and that the stronger inequality |X| ≤ωL(X) ^{ψc (X)t(X)} is true when X is locally H-closed or locally Lindelöf.