Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

Volume 223 / 2013

Samuel Coskey, Tamás Mátrai, Juris Steprāns Fundamenta Mathematicae 223 (2013), 29-48 MSC: 03E15, 03E17. DOI: 10.4064/fm223-1-2


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.


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

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