The Hurewicz covering property and slaloms in the Baire space

Volume 181 / 2004

Boaz Tsaban Fundamenta Mathematicae 181 (2004), 273-280 MSC: Primary 37F20; Secondary 26A03, 03E75. DOI: 10.4064/fm181-3-5


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 ${{\mathfrak b}}$ of an unbounded family of functions in the Baire space. In particular, we obtain a new characterization of ${{\mathfrak b}}$.


  • Boaz TsabanEinstein Institute of Mathematics
    Hebrew University of Jerusalem
    Givat Ram, Jerusalem 91904, Israel

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