Fine ultrafilter
= Fine ultrafilter
{title2=$\{x:\alpha\in x\}\in U$}
An <ultrafilter> on $P_\kappa(\lambda)$ is fine when it contains the set of small subsets containing $\alpha$, for each $\alpha<\lambda$. Fineness makes every coordinate of $\lambda$ represented in the corresponding <ultrapower>, though no one small subset is required to contain every coordinate.