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# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Combinatorics of open covers (VII): Groupability

### Tom 179 / 2003

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

#### Streszczenie

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 ${\mathsf{C}}_{\rm p}(X)$.

#### Autorzy

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

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