On cardinality bounds involving the weak Lindelöf degree

Abstract

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).Separately, it is shown that if X is a regular space with a π-base whose elements have compact closure, then |X| ≤ 2wL(X)ψ(X)t(X). This partially answers a question from [4] and gives a third, separate proof of (1). We also show that if X is a weakly Lindelöf, normal, sequential space with χ(X) ≤ 2ℵ0, then |X| ≤ 2ℵ0.Result (2) above is a new generalization of the cardinality bound 2t(X) for a power homogeneous compactum X (Arhangel'skii, van Mill, and Ridderbos [3], De la Vega in the homogeneous case [10]). To this end we show that if U ⊆ clD ⊆ X, where X is power homogeneous and U is open, then |U| ≤ |D|πχ(X). This is a strengthening of a result of Ridderbos [19].