On cardinality bounds involving the weak Lindelöf degree

Abstract

We give a general closing-off argument in Theorem 2.1 from which several corollaries follow, including (1) if X is a locally compact Hausdorff space then |X| ≤ 2 wL(X)ψ(X) , and (2) if X is a locally compact power homogeneous Hausdorff space then |X| ≤ 2 wL(X)t(X) . The first extends the well-known cardinality bound 2 ψ(X) for a compactum X in a new direction. As |X| ≤ 2 wL(X)χ(X) for a normal space X [3], this enlarges the class of known Tychonoff spaces for which this bound holds. In 2.10 we give a short, direct proof of (1) that does not use 2.1. Yet 2.1 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| ≤ 2 wL(X)χ(X) for all B ∈ B, then |X| ≤ 2 wL(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| ≤ 2 wL(X)ψ(X)t(X) . This partially answers a question from [3] and gives a third, separate proof of (1). We also show that if X is a weakly Lindelöf, normal, sequential space with χ(X) ≤ 2 ℵ0 , then |X| ≤ 2 ℵ0 .Result (2) above is a new generalization of the cardinality bound 2 t(X) for a power homogeneous compactum X (Arhangel'skii, van Mill, and Ridderbos [2], De la Vega in the homogeneous case [9]). 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 [18].2010 Mathematics Subject Classification. 54D20, 54D45, 54A25, 54E99.

On the weak tightness and power homogeneous compacta

“…Lemma 4.5 was shown by Ridderbos in [15]. It was in fact generalized in [5]: if X is power homogeneous, D ⊆ X, and U is an open set such that U ⊆ D, then |U | ≤ |D| πχ(X) . Lemma 4.5 (Ridderbos [15]).…”

Cardinal estimates involving the weak Lindelöf game

“…For example, in [4] Alas proved, among other things, that |X | ≤ 2 wL c (X )•χ(X ) for every Urysohn space X . In [5] Bella and Carlson proved that Question 1.1 has a positive answer for locally compact spaces and in [6] Gotchev proved that Question 1.1 has a positive answer for spaces with a regular G δ diagonal.…”

A survey of cardinality bounds on homogeneous topological spaces

“…Our last improvement of Theorem 6.1 gives a bound for the cardinality of an open set in a power homogeneous space. Theorem 6.6 (Bella, C. [11], 2018). If X is a power homogeneous Hausdorff space and…”

“…The next four results from [45], [11], and [14] represent extensions of de la Vega's Theorem in a different direction using the invariants wL(X) or wL c (X). The weak Lindelöf degree of a space X is the least infinite cardinal κ such that every open cover U of X has a subfamily V such that |V| ≤ κ and X = V. The invariant wL c (X), the weak Lindelöf degree with respect to closed sets, is the smallest infinite cardinal κ such that for every closed subset C of X and every collection U of open sets in X that cover C, there is a subcollection V of U such that |V| ≤ κ and C ⊆ V. It is clear that wL(X) ≤ wL c (X) ≤ aL c (X).…”