  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 (Hurewicz 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.
