Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

Abstract

We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality $\mathfrak{p}\le \mathfrak{b}$ does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting numbers. Lastly, we show that the inclusion ordering on $\mathcal{P}(\omega )$ embeds into the Borel Tukey ordering on cardinal invariants.

References in corpus (1)

• Cofinality spectrum theorems in model theory, set theory and general topology