A connection between decomposable ultrafilters and possible cofinalities. II

We use Shelah's theory of possible cofinalities in order to solve a problem about ultrafilters. THEOREM. Suppose that $\lambda$ is a singular cardinal, $\lambda ' < \lambda$, and the ultrafilter $D$ is $\kappa$-decomposable for all regular cardinals $\kappa$ with $\lambda ' < \kappa < \lambda$. Then $D$ is either $\lambda$-decomposable, or $\lambda ^+$-decomposable. We give applications to topological spaces and to abstract logics.