Cohen algebras and nowhere dense ultrafilters (Q2725204)

Given an ultrafilter \(\xi\) on \(\omega\), there is a natural topological space with underlying set \(\omega^{<\omega}\) which the authors denote \(\text{Seq} (\xi)\). A subset \(U\subset \omega^{<\omega}\) is open if for each \(t\in U\), there is an \(A\in \xi\) such that \(\{ t^\frown n: n\in A\}\) is contained in \(U\). The authors give a nice review of the properties of this space and its Stone-Čech compactification \(\beta \text{ Seq} (\xi)\). In particular, they prove that \(\text{Seq} (\xi)\) and \(\beta\text{ Seq} (\xi)\) are separable extremally disconnected spaces, that nowhere dense subsets of \(\text{Seq} (\xi)\) are scattered, as well as special properties that hold if \(\xi\) is a \(P\)-point or is a weak \(P\)-point. The main thrust of the paper is to determine for which \(\xi\) does \(\beta \text{ Seq} (\xi))\) support either an irreducible map or a semi-open map onto some \(2^\kappa\). By Stone duality and forcing terminology, this is equivalent to asking if the \(\sigma\)-centered clopen algebra on \(\text{Seq} (\xi)\) is forcing equivalent to adding some number of Cohen reals, or if it adds at least one Cohen real. It is shown that, for all \(\xi\), \(\text{Seq} (\xi)\) has a tiny sequence (closely related to the clopen algebra forcing adding a dominating real) and therefore \(\beta\text{ Seq} (\xi)\) does not map irreducibly onto any power of \(2\). Finally, it is shown that \(\beta\text{ Seq} (\xi)\) does map, by a semi-open map, onto \(2^\omega\) exactly when \(\xi\) is not a nowhere dense ultrafilter. \textit{J. E. Baumgartner} [J. Symb. Log. 60, No. 2, 624-639 (1995; Zbl 0834.04005)] defines \(\xi\) to be nowhere dense if for each function \(f\) from \(\omega\) into the reals, there is a member of \(\xi\) whose image is nowhere dense. Baumgartner showed that \(P\)-points are nowhere dense, and Shelah has shown that it is consistent that there are no nowhere dense ultrafilters.