On sequentially Ramsey sets

Anna Brzeska¹

¹ Institute of Mathematics Silesian University Bankowa 14 40-007 Katowice, Poland

Colloquium Mathematicum, Tome 131 (2013) no. 1, pp. 141-148.

Résumé

We consider sequentially completely Ramsey and sequentially nowhere Ramsey sets on $\omega ^\omega $ with the topology generated by a free filter $\mathcal F$ on $\omega $. We prove that if $\mathcal F$ is an ultrafilter, then the $\sigma $-algebra of Baire sets is the $\sigma $-algebra $S_{\mathcal F}\mathcal {CR}$ of sequentially completely Ramsey sets. Further we study additivity and cofinality of the $\sigma $-ideal $S_{\mathcal F}\mathcal {CR}^0$ of sequentially nowhere Ramsey sets. We prove that if $\mathcal F$ is a $P(\mathfrak b)$-ultrafilter then ${\rm add}(S_{\mathcal F}\mathcal {CR}^0)=\mathfrak b$, and if $\mathcal F$ is a $P$-ultrafilter then ${\rm cof}(S_{\mathcal F}\mathcal {CR}^0)$ is the point $\pi $-character of the space $\operatorname {Seq(\mathcal F)}$.

Détail

DOI : 10.4064/cm131-1-12

Keywords: consider sequentially completely ramsey sequentially nowhere ramsey sets omega omega topology generated filter mathcal omega prove mathcal ultrafilter sigma algebra baire sets sigma algebra mathcal mathcal sequentially completely ramsey sets further study additivity cofinality sigma ideal mathcal mathcal sequentially nowhere ramsey sets prove mathcal mathfrak ultrafilter mathcal mathcal mathfrak mathcal p ultrafilter cof mathcal mathcal point character space operatorname seq mathcal