On Cardinality Bounds Involving the Weak Lindelöf Degree and H-Closed Spaces

1. Bella and Carlson give several classes of spaces X for which |X| ≤ 2wL(X)χ(X). This includes locally compact spaces and, more recently, extremally disconnected spaces. Three proofs of the former lead to more general results. One such result is that any regular space X with a π-base consisting of elements with compact closure satisfies |X| ≤ 2wL(X)χ(X). It is also shown that if X is locally compact and power homogeneous that |X| ≤ 2wL(X)t(X), an extension of De la Vega\u27s Theorem. 2. Porter and Carlson give a new cardinality bound for any Hausdorff space that answers a long-standing question of Bella\u27s on H-closed spaces. Using an open ultrafilter assignment, a cardinal invariant L̂(X) is defined with properties a) L̂(X) ≤ L(X), b) L̂(X) is countable if X is H-closed, and c) |X| ≤ 2L̂(X)χ(X) for any Hausdorff space X. This gives a common proof of Arhangel\u27skii\u27s Theorem and the cardinality bound 2χ(X) for H-closed spaces given by Dow and Porter in 1982