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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm223-1-2,
author = {Samuel Coskey and Tam\'as M\'atrai and Juris Stepr\=ans},
title = {Borel Tukey morphisms and combinatorial cardinal invariants of the continuum},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {29-48},
zbl = {06221011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm223-1-2}
}
Samuel Coskey; Tamás Mátrai; Juris Steprāns. Borel Tukey morphisms and combinatorial cardinal invariants of the continuum. Fundamenta Mathematicae, Tome 220 (2013) pp. 29-48. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm223-1-2/