Are All Filters Both Finer and Coarser Than Themselves?

[Saketh]

I am confused by the definition of fineness on filters. Are all filters both finer and coarser than themselves?

 

[andytoh]

A filter is finer than another if it contains it. Just like a topology being finer than another. All filters are contained in a maximal filter, called an ultrafilter. Zorn's Lemma can be used to prove that all filters are contained in an ultrafilter.

All filters both finer and coarser than themselves, but not strictly finer or coarser of course.