Are All Filters Both Finer and Coarser Than Themselves?

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The discussion centers on the concept of fineness in filters, questioning whether all filters can be considered both finer and coarser than themselves. A filter is defined as finer than another if it contains it, similar to the relationship between topologies. It is noted that all filters are contained within a maximal filter known as an ultrafilter, with Zorn's Lemma providing a proof for this containment. The conclusion emphasizes that while filters can be both finer and coarser than themselves, they are not strictly so. This nuanced understanding of filter relationships is crucial in topology and set theory.
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I am confused by the definition of fineness on filters. Are all filters both finer and coarser than themselves?
 
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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.
 

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