According to a result of Kočinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals X has the Hurewicz property if, and only if, each large open cover of X contains a groupable subcover. This solves in the affirmative a problem of Scheepers. The proof uses a rigorously justified abuse of notation and a "structure" counterpart of a combinatorial characterization, in terms of slaloms, of the minimal cardinality 𝔟 of an unbounded family of functions in the Baire space. In particular, we obtain a new characterization of 𝔟.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-3-5,
author = {Boaz Tsaban},
title = {The Hurewicz covering property and slaloms in the Baire space},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {273-280},
zbl = {1056.54028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-3-5}
}
Boaz Tsaban. The Hurewicz covering property and slaloms in the Baire space. Fundamenta Mathematicae, Tome 184 (2004) pp. 273-280. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-3-5/