The combinatorics of open covers (II)

TL;DR

This paper extends the study of diagonalization properties of open covers in separable metrizable spaces, establishing new distinctions, characterizations, and cardinal invariants related to classical properties like Rothberger, Menger, and Hurewicz.

Contribution

It introduces new diagonalization properties, proves their distinctions, characterizes some via classical properties, and determines associated cardinal invariants, also disproving a conjecture of Hurewicz.

Findings

Most new properties are distinct from classical ones.

Certain properties are equivalent to all finite powers having classical properties.

The minimal size of spaces failing these properties equals known small cardinals.

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.