We give a general closing-off argument in Theorem 2.1from which several corollaries follow, including (1) ifXis a locallycompact Hausdorff space then|X| ≤2wL(X)ψ(X), and (2) ifXisa locally compact power homogeneous Hausdorff space then|X| ≤2wL(X)t(X). The first extends the well-known cardinality bound2ψ(X)for a compactumXin a new direction. As|X| ≤2wL(X)χ(X)for a normal spaceX, this enlarges the class of known Tychonoffspaces for which this bound holds. In 2.10 we give a short, directproof of (1) that does not use 2.1. Yet 2.1 is broad enough toestablish results much more general than (1), such as ifXis aregular space with aπ-baseBsuch that|B| ≤2wL(X)χ(X)for allB∈B, then|X| ≤2wL(X)χ(X).Separately, it is shown that ifXis a regular space with aπ-basewhose elements have compact closure, then|X| ≤2wL(X)ψ(X)t(X).This partially answers a question from  and gives a third, sep-arate proof of (1). We also show that ifXis a weakly Lindel ̈of,normal, sequential space withχ(X)≤2א0, then|X| ≤2א0.Result (2) above is a new generalization of the cardinality bound2t(X)for a power homogeneous compactumX(Arhangel’skii, vanMill, and Ridderbos , De la Vega in the homogeneous case ).To this end we show that ifU⊆clD⊆X, whereXis power ho-mogeneous andUis open, then|U| ≤ |D|πχ(X). This is a strength-ening of a result of Ridderbos .
|Titolo:||On cardinality bounds involving the weak Lindelöf degree|
BELLA, Angelo (Corresponding)
|Data di pubblicazione:||2018|
|Appare nelle tipologie:||1.1 Articolo in rivista|
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