The weak Lindelöf degree and homogeneity

In this note a unifying closing-off argument is given (Theorem 2.5) involving the weak Lindelöf degree wL(X) of a Hausdorff space X and covers of X by compact subsets. This has among it corollaries the known cardinality bound 2wLc(X)χ(X) for spaces with Urysohn-like properties (Alas, 1993, [1], , Bonanzinga, Cammaroto, and Matveev, 2011, [9], ), the known bound cardinality bound 2wL(X)χ(X) for spaces with a dense set of isolated points (Dow and Porter, 1982, [14], ), and two new cardinality bounds for power homogeneous spaces. In particular, it is shown that (a) if X is a power homogeneous Hausdorff space that is either quasiregular or Urysohn, then |X|⩽2wLc(X)t(X)pct(X), and (b) if X is a power homogeneous Hausdorff space with a dense set of isolated points then |X|⩽2wL(X)t(X)pct(X). These two bounds represent improvements on bounds for power homogeneous spaces given in Carlson et al. (2012) [11], as wLc(X)⩽aLc(X) for any space X. These results establish that known cardinality bounds for spaces with Urysohn-like properties, as well as spaces with a dense set of isolated points, are consequences of more general results that also give “companion” bounds for power homogeneous spaces with these properties.