Two real ultrafilters on ω

We prove the following results concerning the space ω∪{p}, where p is an ultrafilter on ω: ere are ultrafilters p on ω with the property that every compactification of ω∪{p} contains a copy of βω. here is an ultrafilter p on ω with a strong Noetherian base, i.e., every member of the base is contained, mod finite, in only finitely many others.