Title: The combinatorics of open covers (II)
Authors: Winfried Just, Arnold W. Miller, Marion Scheepers
and Paul J. Szeptycki
Abstract:
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.
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