On cardinality bounds for homogeneous spaces and the -modification of a space

Improving a result in Carlson and Ridderbos (2012) [9], we construct a closing-off argument showing that the Lindelöf degree of the G κ -modification of a space X is at most 2 L ( X ) F ( X ) ⋅ κ , where F ( X ) is the supremum of the lengths of all free sequences in X and κ is an infinite cardinal. From this general result follow two corollaries: (1) | X | ⩽ 2 L ( X ) F ( X ) pct ( X ) for any power homogeneous Hausdorff space X, where pct ( X ) is the point-wise compactness type of X, and (2) | X | ⩽ 2 L ( X ) F ( X ) ψ ( X ) for any Hausdorff space X, as shown recently by Juhász and Spadaro (preprint) [17]. By considering the Lindelöf degree of the related G κ c -modification of a space X, we also obtain two consequences: (1) if X is a power homogeneous Hausdorff space then | X | ⩽ 2 a L c ( X ) t ( X ) pct ( X ) , where a L c ( X ) is the almost Lindelöf degree with respect to closed sets, and (2) | X | ⩽ 2 a L c ( X ) t ( X ) ψ c ( X ) for any Hausdorff space X, a well-known result of Bella and Cammaroto (1988) [4]. This demonstrates that both the Juhász–Spadaro and Bella–Cammaroto cardinality bounds for Hausdorff spaces are consequences of more general results that additionally lead to companion bounds for power homogeneous Hausdorff spaces. Finally, we give cardinality bounds for θ-homogeneous spaces that generalize those for homogeneous spaces, including cases in which the Hausdorff condition is relaxed.