- The Fréchet Filter:
smallest non-principal ultrafilter. - The family of supersets of $\{1,2\}\subseteq\mathbb{R}$: smallest filter containing $\{1,2\}$. Principal. Not an ultrafilter.
- Any ultrafilter over a finite set is principal.
Are there filters that are neither principal nor an ultrafilter?
An example would be much appreciated. Thanks!
Best Answer
The Fréchet filter $F$ isn't the smallest non-principal ultrafilter. It's the filter consisting of all cofinite sets. And it's not an ultrafilter, since any infinite/co-infinite $X$ has $X\notin F$ and $X^c\notin F.$ As such, it's an excellent example of a filter that is not an ultrafilter and is not principal.