# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

### Tom 223 / 2013

Fundamenta Mathematicae 223 (2013), 29-48 MSC: 03E15, 03E17. DOI: 10.4064/fm223-1-2

#### Streszczenie

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.

#### Autorzy

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