We continue to investigate various diagonalization properties for sequences of open covers of separable metrizable spaces introduced in Part I. These properties generalize classical ones of Rothberger, Menger, Hurewicz, and Gerlits-Nagy. In particular, we show that most of the properties introduced in Part I are indeed distinct. We characterize two of the new properties by showing that they are equivalent to saying all finite powers have one of the classical properties above (Rothberger property in one case and in the Menger property in other). We consider for each property the smallest cardinality of metric space which fails to have that property. In each case this cardinal turns out to equal another well-known cardinal less than the continuum. We also disprove (in ZFC) a conjecture of Hurewicz which is analogous to the Borel conjecture. Finally, we answer several questions from Part I concerning partition properties of covers. 3

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A space X is said to have property (USC) (resp. (LSC)) if whenever {fn : n ∈ ω} is a sequence of upper (resp. lower) semicontinuous functions from X into the closed unit interval [0, 1] converging pointwise to the constant function 0 with the value 0, there is a sequence {gn : n ∈ ω} of continuous functions from X into [0, 1] such that fn ∑ gn (n ∈ ω) and {gn : n ∈ ω} converges pointwise to 0. In this paper, we study spaces having these properties and related ones. In particular, we show that (a) for a subset X of the real line, X has property (USC) if and only if it is a σ-set; (b) if X is a space of non-measurable cardinal and has property (LSC), then it is discrete. Our research comes of Scheepers' conjecture on properties S 1 (Γ, Γ) and wQN.

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