I still don't understand how is that true. Here are the definitions: Let P be a class of closed subsets of a topological space X which is closed under finite intersections and finite unions. A closed filter F is a collection of non-empty elements of P with the properties: 1. F is closed under finite intersections 2. If A is in F and B is a superset of A belonging to P, then B belongs to F.
A closed ultrafilter is a closed filter F satisfying: Whenever A belongs to P and A meets each element of F, then A is in F.
A prime closed filter is a closed filter F satisfying : If A and B belong to P and their union belongs to F, then either A is in F or B is in F.
I don't understand why a prime closed filter can't be a closed ultrafilter, yet a closed ultrafilter is always a prime closed filter.