Weak Covering Properties of Weak Topologies

Abstract

We consider covering properties of weak topologies of Banach spaces, especially of weak or point-wise topologies of function spaces C(K), for compact spaces K. We answer questions posed by A. V. Arkhangel'skii, S. P. Gul'ko, and R. W. Hansell. Our main results are the following. A Banach space of density at most ω1 is hereditarily metaLindel of in its weak topology. If the weight of a compact spaceK is at most ω1, then the spaces Cw(K) and Cp(K) are hereditarily metaLindel of. Let T ¯ be the one-point compactification of a treeT. Then the space C p ( T ¯ ) is hereditarily σ-metacompact. If T is an infinitely branching full tree of uncountable height and of cardinality bigger than c, then the weak topology of the unit sphere of C ( T ¯ ) is not σ-fragmented by any metric. The space Cp(rβω1) is neither metaLindel of nor σ-relatively metacompact. The space Cp(rβω2) is not σ-relatively metaLindel of. Under the set-theoretic axiom ♦, there exists a scattered compact space K1 such that the space Cp(K1) is not σ-relatively metacompact, and under a related axiom ◊, there exists a scattere compact space K2 such that the space Cp(K2) is not σ-relatively metaLindel of. 1991 Mathematics Subject Classification: 54C35, 46B20, 54E20, 54D30.