Combinatorics of open covers (VII): Groupability

Tom 179 / 2003

Ljubiša D. R. Kočinac, Marion Scheepers Fundamenta Mathematicae 179 (2003), 131-155 MSC: 54D20, 54C35, 54A25, 03E02, 91A44. DOI: 10.4064/fm179-2-2


We use Ramseyan partition relations to characterize:

$\bullet$ the classical covering property of Hurewicz;

$\bullet$ the covering property of Gerlits and Nagy;

$\bullet$ the combinatorial cardinal numbers $\mathfrak{b}$ and $\mathsf{add}({\mathcal M})$.

Let $X$ be a $\mathsf T_{3\frac{1}{2}}$-space. In \cite{KS2} we showed that ${\mathsf{C}}_{\rm p}(X)$ has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of $X$ have the Gerlits–Nagy covering property. Now we show that the following are equivalent:

1. ${\mathsf{C}}_{\rm p}(X)$ has countable fan tightness and the Reznichenko property.

2. All finite powers of $X$ have the Hurewicz property.

We show that for ${\mathsf{C}}_{\rm p}(X)$ the combination of countable fan tightness with the Reznichenko property is characterized by a Ramseyan partition relation. Extending the work in \cite{KS2}, we give an analogous Ramseyan characterization for the combination of countable strong fan tightness with the Reznichenko property on ${\mathsf{C}}_{\rm p}(X)$.


  • Ljubiša D. R. KočinacDepartment of Mathematics
    Faculty of Sciences
    University of Niš
    18000 Niš, Yugoslavia
  • Marion ScheepersDepartment of Mathematics
    Boise State University
    Boise, ID 83725, U.S.A.

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