Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

Samuel Coskey¹ ; Tamás Mátrai² ; Juris Steprāns³

¹ Department of Mathematics Boise State University 1910 University Dr. Boise, ID 83725-1555, U.S.A. Formerly at York University Toronto, Canada
² Alfréd Rényi Matematikai Kutatóintézet Magyar Tudományos Akadémia 13-15 Reáltanoda utca H-1053 Budapest, Hungary
³ Department of Mathematics and Statistics, N520 Ross York University 4700 Keele St. Toronto, ON, M3J 1P3, Canada

Fundamenta Mathematicae, Tome 223 (2013) no. 1, pp. 29-48.

Résumé

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.

Détail

DOI : 10.4064/fm223-1-2

Keywords: discuss borel tukey ordering cardinal invariants continuum observe ordering makes sense larger class cardinals has previously considered provide borel version large portion van nbsp douwens diagram instance although usual proof inequality mathfrak leq mathfrak does provide borel tukey map there afterwards revisit result mildenberger concerning generalization unsplitting splitting numbers lastly results embedding inclusion ordering mathcal omega borel tukey ordering cardinal invariants