Limits in the uniform ultrafilters

Abstract

Let $u(\kappa )$ be the space of uniform ultrafilters on $\kappa$. If $\kappa$ is regular, then there is an $\mathbf x \in u(\kappa )$ which is not an accumulation point of any subset of $u(\kappa )$ of size $\kappa$ or less. $\mathbf x$ is also good, in the sense of Keisler.