On cardinality bounds involving the weak Lindelöf degree

Published in: Quaestiones Mathematicae
Volume 41, issue 1, 2018, pages: 99–113
DOI: 10.2989/16073606.2017.1373157
Author(s): A. BellaDepartment of Mathematics, Italy, N. CarlsonDepartment of Mathematics, USA


We give a general closing-off argument in Theorem 2.3 from which several corollaries follow, including (1) if X is a locally compact Hausdorff space then |X| ≤ 2wL(X)ψ(X), and (2) if X is a locally compact power homogeneous Hausdorff space then |X| ≤ 2wL(X)t(X). The first extends the well-known cardinality bound 2ψ(X) for a compactum X in a new direction. As |X| ≤ 2wL(X)χ(X) for a normal spaceX[4], this enlarges the class of known Tychonoff spaces for which this bound holds. In 2.12 we give a short, direct proof of (1) that does not use 2.3. Yet 2.3 is broad enough to establish results much more general than (1), such as if X is a regular space with a π-base ℬ such that |B| ≤ 2wL(X)χ(X) for all B ∈ ℬ, then |X| ≤ 2wL(X)χ(X).

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