Weak Covering Properties of Weak Topologies

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 ω 1 is hereditarily metaLindel of in its weak topology. If the weight of a compact space K is at most ω 1 , then the spaces C w ( K ) and C p ( K ) are hereditarily metaLindel of. LetT ¯be the one‐point compactification of a tree T . 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 C p (rβω 1 ) is neither metaLindel of nor σ‐relatively metacompact. The space C p (rβω 2 ) is not σ‐relatively metaLindel of. Under the set‐theoretic axiom ♦, there exists a scattered compact space K 1 such that the space C p ( K 1 ) is not σ‐relatively metacompact, and under a related axiom ◊, there exists a scattere compact space K 2 such that the space C p ( K 2 ) is not σ‐relatively metaLindel of. 1991 Mathematics Subject Classification : 54C35, 46B20, 54E20, 54D30.