The Hurewicz covering property and slaloms in the Baire space

Boaz Tsaban¹

¹ Einstein Institute of Mathematics Hebrew University of Jerusalem Givat Ram, Jerusalem 91904, Israel

Fundamenta Mathematicae, Tome 181 (2004) no. 3, pp. 273-280

Résumé

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 ${{\mathfrak b}}$.

Détail

DOI : 10.4064/fm181-3-5

Keywords: according result inac scheepers hurewicz covering property equivalent somewhat simpler selection property each sequence large covers space choose finitely many elements each cover obtain groupable cover space simplify characterization further omitting consider sequences covers set reals has hurewicz property only each large cover contains groupable subcover solves affirmative problem scheepers proof uses rigorously justified abuse notation structure counterpart combinatorial characterization terms slaloms minimal cardinality mathfrak unbounded family functions baire space particular obtain characterization mathfrak