A Small Ultrafilter Number at Every Singular Cardinal

Abstract

We obtain a small ultrafilter number at ${\aleph }_{{\omega }_1}$. Moreover, we develop a version of the overlapping strong extender forcing with collapses which can keep the top cardinal $\kappa$ inaccessible. We apply this forcing to construct a model where $\kappa$ is the least inaccessible and $V_{\kappa }$ is a model of GCH at regulars, failures of SCH at singulars, and the ultrafilter numbers at all singulars are small.