The growth of subuniform ultrafilters

Abstract:

Some of the results on the topology of spaces of uniform ultrafilters are applied to the space $\Omega ({\alpha ^ + })$ of subuniform ultrafilters (i.e., the set of ultrafilters which are $\alpha$-uniform but not ${\alpha ^ + }$-uniform) on ${\alpha ^ + }$ when $\alpha$ is a regular cardinal. The main object is to find for infinite cardinals $\alpha$, such that $\alpha = {\alpha ^{\underline {a}}}$, a topological property that separates the space $\beta (\Omega ({\alpha ^ + }))\backslash \Omega ({\alpha ^ + })$ (the growth of $\Omega ({\alpha ^ + })$) from the space $U({\alpha ^ + })$ of uniform ultrafilters on ${\alpha ^ + }$. Property ${\Phi _\alpha }$ fulfils this rôle defined for a zero-dimensional space X by the following condition: every nonempty closed subset of X of type at most $\alpha$ is not contained in the uniform closure of a family of $\alpha$ pairwise disjoint nonempty open-and-closed subsets of X. The “infinitary” properties of $\Omega ({\alpha ^ + })$, as they are measured by ${\Phi _\alpha }$, are more closely related to those of $U(\alpha )$ than to those of $U({\alpha ^ + })$. A consequence of this topological separation is that the growth of $\Omega ({\alpha ^ + })$ is not homeomorphic to $U({\alpha ^ + })$ and, in particular, that $\Omega ({\alpha ^ + })$ is not ${C^ \ast }$-embedded in the space $\Sigma ({\alpha ^ + })$ of $\alpha$-uniform ultrafilters on ${\alpha ^ + }$. These results are related to, and imply easily, the Aronszajn-Specker theorem: if $\alpha = \alpha ^{\underline {a}}$ then ${\alpha ^ + }$ is not a ramifiable cardinal. It seems possible that similar questions on the ${C^ \ast }$-embedding of certain spaces of ultrafilters depend on (and imply) results in partition calculus.