2021/10/25 by Bella, Angelo, Carlson, Nathan, Gotchev, Ivan
#54A25 #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2110.13122
We give several new bounds for the cardinality of a Hausdorff topological space X involving the weak Lindelöf degree wL(X). In particular, we show that if X is extremally disconnected, then |X|≤ 2wL(X)πχ(X)ψ(X), and if X is additionally power homogeneous, then |X|≤ 2wL(X)πχ(X). We also prove that if X is an almost Lindelöf space with a strong Gδ-diagonal of rank 2, then |X|≤ 2ℵ0; that if X is a star-cdc space with a Gδ-diagonal of rank 3, then |X| ≤ 2ℵ0; and if X is any normal star-cdc space X with a Gδ-diagonal of rank 2, then |X|≤ 2ℵ0. Several improvements of results in [9] 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|≤ 2ψc(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 Lindelöf.