On sequentially Ramsey sets

Volume 131 / 2013

Anna Brzeska Colloquium Mathematicum 131 (2013), 141-148 MSC: Primary 54A20, 54D35; Secondary 54A25, 54D70, 54D80. DOI: 10.4064/cm131-1-12


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 $\operatorname {Seq(\mathcal F)}$.


  • Anna BrzeskaInstitute of Mathematics
    Silesian University
    Bankowa 14
    40-007 Katowice, Poland

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