Combinatorics of open covers (VII): Groupability

Abstract

We use Ramseyan partition relations to characterize: the classical covering property of Hurewicz; the covering property of Gerlits and Nagy; the combinatorial cardinal numbers b and add(script M sign). Let X be a T 31/2-space. In [9] we showed that Cp(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. Cp(X) has countable fan tightness and the Reznichenko property. 2. All finite powers of X have the Hurewicz property. We show that for Cp(X) the combination of countable fan tightness with the Reznichenko property is characterized by a Ramseyan partition relation. Extending the work in [9], we give an analogous Ramseyan characterization for the combination of countable strong fan tightness with the Reznichenko property on Cp(X).