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

摘要

Improving a result in Carlson and Ridderbos (2012) [9], we construct a dosing-off argument showing that the Lindelof degree of the G_K -modification of a space X is at most 2~(L(X)F(X).K), where F(X) is the supremum of the lengths of all free sequences in X and k 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 Juhasz and Spadaro (preprint) [17]. By considering the Lindelof degree of the related G_K~C -modification of a space X, we also obtain two consequences: (1) if X is a power homogeneous Hausdorff space then |X|≤2~(L(X)F(X)pct(X)), where aL_c(X) is the almost Lindelof degree with respect to closed sets, and (2) |X| ≤2~(L(X)F(X)Ψ(X)) for any Hausdorff space X, a well-known result of Bella and Cammaroto (1988) [4]. This demonstrates that both the Juhasz-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 fl-homogeneous spaces that generalize those for homogeneous spaces, including cases in which the Hausdorff condition is relaxed.