Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Colloquium Mathematicum / All issues

Abstract

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