Borel Tukey morphisms and combinatorial cardinal invariants of the continuum
Volume 223 / 2013
Fundamenta Mathematicae 223 (2013), 29-48
MSC: 03E15, 03E17.
DOI: 10.4064/fm223-1-2
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 ${\mathfrak {p}}\leq {\mathfrak {b}}$ 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 use our results to give an embedding from the inclusion ordering on $\mathcal P(\omega )$ into the Borel Tukey ordering on cardinal invariants.
