Borel Tukey Morphisms and Combinatorial Cardinal Invariants of the Continuum
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 p≤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 P(ω) into the Borel Tukey ordering on cardinal invariants.
Coskey, Samuel; Mátrai, Tamás; and Steprāns, Juris. (2013). "Borel Tukey Morphisms and Combinatorial Cardinal Invariants of the Continuum". Fundamenta Mathematicae, 223(1), 29-48.