arXiv math/03010852003-01math.GNmath.COmath.LO

The Hurewicz covering property and slaloms in the Baire space

Boaz Tsaban

备注：: Small updates
发表信息：: Fundamenta Mathematicae 181 (2004), 273--280
DOI：: 10.4064/fm181-3-5

英文摘要：

According to a result of Kocinac 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 $X$ satisfies 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 $\b$.

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