The weak Lindelof degree and homogeneity

In this note a unifying closing-off argument is given (Theorem 2.5) involving the weak Lindelof degree wL(X) of a Hausdorff space X and covers of X by compact subsets. This has among it corollaries the known cardinality bound 2(wLc(X)chi(X)) for spaces with Urysohn-like properties (Alas, 1993, [1], Bonanzinga, Cammaroto, and Matveev, 2011, [9]), the known bound cardinality bound 2(wL(X)chi(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 vertical bar X vertical bar <= 2(wLc(X)t(X)pct(X)), and (b) if X is a power homogeneous Hausdorff space with a dense set of isolated points then vertical bar X vertical bar <= 2(wL(X)t(X)pct(X)). These two bounds represent improvements on bounds for power homogeneous spaces given in Carlson et al. (2012) [11], as wL(c)(X) <= aL(c)(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. (C) 2013 Elsevier B.V. All rights reserved.